  We report first-principles all-electron density-functional based studies of
the electronic structure, magnetic ordering and anisotropy for the V$_{15}$
molecular magnet. From these calculations, we determine a Heisenberg
Hamiltonian with four antiferromagnetic and one {\em ferromagnetic} coupling.
We perform direct diagonalization to determine the temperature dependence of
the susceptibility. This Hamiltonian reproduces the experimentally observed
spin $S$=1/2 ground state and low-lying $S$=3/2 excited state. A small
anisotropy term is necessary to account for the temperature independent part of
the magnetization curve.
