The flow produced by slowly squeezing a yield stress fluid between two circular, parallel plates is analyzed using a variational principle. Previous anayses of this problem have been largely unsuccessful because they have relied on direct solution methods which require the complete determination of the flow field. Although a complete determination of the flow field is a desirable objective, it is not necessary for obtaining upper and lower bounds for the rate of squeezing of the yield stress fluid. Here we have developed such bounds for the squeezing flow of a Herschel-Bulk-ley fluid. The lower bound is found to be similar to the previously reported solutions. We have also carried out some related experiments. These experimental results are in agreement with the form of the lower bound to the solution.