We prove that for $k+1\geq 3$ and $c>(k+1)/2$ w.h.p. the random graph on $n$ vertices, $cn$ edges and minimum degree $k+1$ contains a (near) perfect $k$-matching. As an immediate consequence we get that w.h.p. the $(k+1)$-core of $G_{n,p}$, if non empty, spans a (near) spanning $k$-regular subgraph. This improves upon a result of Chan and Molloy and completely resolves a conjecture of Bollobás, Kim and Verstraëte. In addition, we show that w.h.p. such a subgraph can be found in linear time. A substantial element of the proof is the analysis of a randomized algorithm for finding $k$-matchings in random graphs with minimum degree $k+1$.