Recently, sharply reduced magnetic moments were measured in Mn13 and Mn19 clusters with values of about 0.5 $\mu$B/atom, while other clusters had moments of 1 to 1.4$\mu$B/atom. It was postulated that this sharply reduced magnetic moment results from the icosahedral growth sequence. We confirm the stability of the icosahedral structures. The icosahedral Mn13 and Mn19 clusters have several nearly degenerate low energy states, each with a low magnetic moment. It is shown that the mixture of ferromagnetic and antiferromagnetic coupling is responsible for the low net magnetic moments.