Manski (1985) has shown that the maximum score estimator of the coefficient vector of a binary response model is consistent under weak distributional assumptions. Cavanagh (1987) and Kim and Pollard (1989) have shown that $N^{1/3}$ times the centered maximum score estimator converges in distribution to the random variable that maximizes a certain Gaussian process. The properties of the limiting distribution are largely unknown, and the result of Cavanagh and Kim and Pollard cannot be used for inference in applications. This paper describes a modified maximum score estimator that is obtained by maximizing a smoothed version of Manski's score function. Under distributional assumptions that are somewhat stronger than Manski's but still very weak, the centered smoothed estimator is asymptotically normal with a convergence rate that is at least $N^{-2/5}$ and can be made arbitrarily close to $N^{-1/2}$, depending on the strength of certain smoothness assumptions. The estimator's rate of convergence is the fastest possible under the assumptions that are made. The parameters of the limiting distribution can be estimated consistently from data, thereby making statistical inference based on the smoothed estimator possible with samples that are sufficiently large.