We study the {\em robust proper learning} of univariate log-concave distributions (over continuous and discrete domains). Given a set of samples drawn from an unknown target distribution, we want to compute a log-concave hypothesis distribution that is as close as possible to the target, in total variation distance. In this work, we give the first computationally efficient algorithm for this learning problem. Our algorithm achieves the information-theoretically optimal sample size (up to a constant factor), runs in polynomial time, and is robust to model misspecification with nearly-optimal error guarantees. Specifically, we give an algorithm that, on input $n=O(1/\eps^{5/2})$ samples from an unknown distribution $f$, runs in time $\widetilde{O}(n^{8/5})$, and outputs a log-concave hypothesis $h$ that (with high probability) satisfies $\dtv(h, f) = O(\opt)+\eps$, where $\opt$ is the minimum total variation distance between $f$ and the class of log-concave distributions. Our approach to the robust proper learning problem is quite flexible and may be applicable to many other univariate distribution families.