Consider a family of distributions indexed by a parameter \beta, where the probability that the configuration x is chosen is proportional to exp(-\beta H(x)) / Z(\beta). Here Z(\beta) is the proper normalizing constant, equal to the sum over x' of exp(-\beta H(x')). Then {\pi_\beta} is known as a Gibbs distribution, and Z(\beta) is the partition function. This work presents a new method for approximating the partition function to a specified level of relative accuracy using only a number of samples that is O(ln(Z(\beta)) ln(ln(Z(\beta)))) when Z(0) >= 1. This is a sharp improvement over previous similar approaches, which used a much more complicated algorithm requiring O(ln(Z(\beta)) ln(ln(Z(\beta)))^5) samples.