PATRICK E. HIGGINBOTHAM, Circuit Judge:

Between 1981 and 1984, Dixie Machine Welding Metal Works was a party to a multi-employer contract with the International Brotherhood of Boilermakers. Dixie Machine and the union negotiated a separate contract for the 1984 to 1987 period. A dispute developed about overtime rates under the new contract. The parties submitted the dispute to binding arbitration. The arbitrator ruled for the union. Dixie Machine sued in federal district court to vacate the arbitrator's award. The court granted summary judgment for Dixie Machine, and the union appealed. We now reverse and remand to the district court for enforcement of the arbitrator's judgment.

I

The dispute between Dixie Machine and the union deals with the relation between overtime pay rates and health-benefit plan contributions. The 1981-84 ("old") contract contained the following provisions:

(1) Appendix A which listed hourly rates payable to particular workers, subject to the deductions listed in Appendices C and D.

(2) Appendix C, specifying that "contributions will be paid to the Boilermakers National Health Welfare Fund, on the basis of $1.275 per hour, straight time hours only, in any one pay period by all employees covered by this agreement."

(3) Article VIII, Section 6, reading "All overtime shall be paid at the rate of double time . . . of the rates established in this contract, however, on the premium hours $1.275 shall be added because of the Health Welfare Plan."

The corresponding provisions of the 1984-87 ("new") contract were as follows:

(1) Appendix A which listed hourly rates payable to particular workers, but did not make any reference to deductions elsewhere set forth. The rates in the new contract's Appendix A were lower than those in the old contract's Appendix A.

(2) Appendix B, specifying that "contributions will be paid to the Boilermakers National Health Welfare Fund, on the basis of $1.85 per hour, straight time hours only, in any one pay period by all employees covered by this agreement."

(3) Article VIII, specifying overtime rates without reference to the Health Welfare Plan.

(4) A clause at the conclusion of the contract, labeled "For Appendix `B' and Appendix `C'," and stating, "Deduction amounts may be changed from time to time by the following method . . ."

The Company wishes to compute overtime pay rates by applying a multiplier to the rates in Appendix A. The Union contends that the pay rates thus computed must be supplemented by the amount that the Company pays into the benefit plan during straight time, but not overtime, hours, as was done under the old contract.

The arbitrator found for the Union, holding that the parties had a "past practice" by which the benefit contributions would "revert back to the employees" during overtime hours. The arbitrator found that termination of this practice had not been bargained over or even discussed during negotiation of the new contract. He further found that the mere deletion of the language in Article VIII referring to the Health Welfare Plan contributions was not sufficient to terminate the past practice.

Dixie Machine filed suit to vacate the award pursuant to Section 301 of the Labor Management Relations Act, 29 U.S.C. § 185. The union answered and counterclaimed for enforcement of the award. Both sides sought summary judgment. The district court granted summary judgment for Dixie Machine on the ground that the arbitrator's judgment was inconsistent with the "clear and unambiguous" language of the new contract.

II

Federal law severely circumscribes the power of federal courts to review the decisions of labor arbitrators. "As long as the arbitrator's award `draws its essence from the collective bargaining agreement,' and is not merely `his own brand of industrial justice,' the award is legitimate." United Paperworkers Intern. Union v. Misco, Inc., 484 U.S. 28, 108 S.Ct. 364, 370, 98 L.Ed.2d 286 (1987). When "only improvident, even silly, factfinding is claimed," that "is hardly sufficient basis for disregarding what the agent appointed by the parties determined to be the historical facts." 108 S.Ct. at 371. "The arbitrator may not ignore the plain language of the contract, but the parties having authorized the arbitrator to give meaning to the language of the agreement, a court should not reject an award on the ground that the arbitrator misread the contract." 108 S.Ct. at 371. See also Mansville Forest Prod. v. United Paperworkers Intern., 831 F.2d 72, 74 (5th Cir. 1987) (arbitrator's award will be enforced unless "it stems from fraud or partiality; . . . it concerns a matter not subject to arbitration . . .; . . . it does not `dra[w] its essence' from the contract; or . . . it violates public policy").

The district court found that the arbitrator's judgment contradicted the "clear and unambiguous" language of the new contract. However, the court's disagreement with the arbitrator appears to rest more squarely upon a rejection of the arbitrator's factual finding that there existed an established past practice according to which health benefits would revert to the employees. In particular, both the district court and Dixie Machine contest the arbitrator's statement that "[n]either the old . . . Contract, nor the new . . . Agreement, contained language which stated that health and welfare contributions would revert back to employees for overtime hours worked."

Both the district court and Dixie Machine point to the language in Article VIII, Section 6 of the old contract. Yet this language was the subject of much disagreement at the arbitration hearing. The union and Dixie Machine had differing views about how overtime pay rates had been calculated under the old contract. The union contended that the health benefit amount had been added to twice the straight rate, while the company contended it had added the benefit amount to the straight rate before doubling. The language in Article VIII of the old contract fails to make clear whether the extra $1.275 will be added in before or after doubling of the base rate. Moreover, Dixie Machine itself testified that under the new contract it had made health benefit contributions during overtime hours, although the payments were made to the fund rather than to the workers and were, the company claimed, made by mistake.

In light of the very real controversy about what had actually been done under the old contract, the arbitrator was certainly free to exercise his judgment in deciding what the old contract meant. He likewise could determine the significance of the differences between the old and the new contract.

The arbitrator's findings are not obviously "silly," nor do they rest on an obvious "misreading" of the old or the new contract. But even if they were obviously silly, or obviously the result of a misreading of the contract, that would not permit a reviewing court to vacate his determination. Misco, 108 S.Ct. at 370-71.

We therefore need not decide whether the meaning of the old contract is comparable to other "historical facts," or whether the arbitrator's award must "draw its essence from" the old, as well as the new, contract.

Federal law gives very great deference to private ordering in arbitration cases. The parties have contracted to be bound, and bound they stay. Federal courts may set aside an award if the arbitrator acted fraudulently or with evident bias, but there is no allegation of bias or fraud here. Federal courts may also set aside an award if the arbitrator ignored the clear language of the contract. Dixie Machine successfully urged the district court to act on this ground. But the arbitrator clearly did not ignore the relevant language: he quoted it, and commented upon its meaning. Even if Dixie Machine's argument is believed, the company's argument establishes only that the arbitrator misunderstood the language — not that he ignored it. That conclusion does not authorize us to release Dixie Machine from a judgment it agreed to honor.

The district court's grant of summary judgment for Dixie Machine is therefore reversed. We direct summary judgment for the union. We remand the case to the district court for entry of an appropriate order enforcing the arbitrator's award.

REVERSED AND REMANDED.