The positive semidefinite Procrustes (PSDP) problem is the following: given rectangular matrices $X$ and $B$, find the symmetric positive semidefinite matrix $A$ that minimizes the Frobenius norm of $AX-B$. No general procedure is known that gives an exact solution. In this paper, we present a semi-analytical approach to solve the PSDP problem. First, we characterize completely the set of optimal solutions and identify the cases when the infimum is not attained. This characterization requires the unique optimal solution of a smaller PSDP problem where $B$ is square and $X$ is diagonal with positive diagonal elements. Second, we propose a very efficient strategy to solve the PSDP problem, combining the semi-analytical approach, a new initialization strategy and the fast gradient method. We illustrate the effectiveness of the new approach, which is guaranteed to converge linearly, compared to state-of-the-art methods.