Let $G$ be uniformly distributed on the set of all simple $d$-regular graphs on $n$ vertices, and assume $d$ is bigger than some (small) power of $n$. We show that the second largest eigenvalue of $G$ is of order $\sqrt{d}$ with probability close to one. Combined with earlier results covering the case of sparse random graphs, this settles the problem of estimating the magnitude of the second eigenvalue, up to a multiplicative constant, for all values of $n$ and $d$, confirming a conjecture of Van Vu. Joint work with Pierre Youssef.