Specific heat of helium confined to a 57- &mgr;m planar geometry near the lambda point

We report measurements of the specific heat of liquid helium confined to 57-&mgr;m planar gaps extending to within a few nanokelvin of the bulk lambda transition. The data are in fair agreement with Monte Carlo estimates for finite-size effects and with renormalization-group-theory predictions above the transition. Far from the transition, we find surface specific heat exponents, alpha(s) = 0.64+/-0.05 below, and 0.65+/-0.2 above, which compare well with the prediction of 0.658. Comparison with other recent data on small length scales shows some areas of agreement.