Given a symmetric matrix \(G\), one has to find a correlation matrix \(X\) (i.e., a positive semidefinite and symmetric matrix with diagonal elements equal to 1) which is closest to \(G\) in Frobenius norm. The dual of this constrained optimization problem is an unconstrained problem with an objective function that is not continuously differentiable. Hence a direct approach would result in at most linear convergence. However, its solution is obtained by solving a system of nonlinear equations involving the Frobenius norm of a projection onto the cone of positive semidefinite symmetric matrices. Since this projection is strongly semismooth, the authors succeed in designing a (nonsmooth) Newton method which they prove to be quadratically convergent which is confirmed by numerical experiments. Extensions are discussed which include minimizing a weighted Frobenius norm, imposing a lower bound on \(X\), and allowing \(X\) to be nonsymmetric.