We compute the average partition function for an electron moving in a Gaussian random potential. A path integral formulation is used, with a trial action like that in Feynman's polaron theory. We compute the variational bound as well as the first correction in a systematic cumulant expansion. The results are checked against exact formulas for the onedimensional white noise problem. The density of states in the low-energy tail has the correct exponential energy dependence, and energy-dependent prefactor to within a few percent. In addition, the partition function goes over smoothly to the perturbation theory result at high temperatures.