The magnetic properties of 13-atom M clusters (M=Y, Zr, Nb, Mo, and Tc) with three possible highsymmetrygeometries have been studied using the discrete-variational local-spin-density-functional method. While the ground states of most transition-metal 13-atom clusters correspond to the icosahedral structure, we found that the cuboctahedral structure is more energetically stable than the icosahedral one for Mo13 and Tc13 clusters. The ground states of all the clusters are shown to be magnetic, but their magnetic moments are not striking.