We show that there are large regions of parameter space for the supersymmetric standard model in which the lightest supersymmetric particle (LSP) is a nearly pure bino (partner of the hypercharge gauge boson) or particular higgsino (symmetric or antisymmetric linear combination of the two neutral higgsinos with definite hypercharges). In particular, a very massive LSP is almost always one of these possibilities. We calculate the total annihilation cross section for these LSPs, including gauge and Higgs bosons as final states. By computing the relic abundance of LSPs left over from the Big Bang, we can rule out some portions of the parameter space. In other portions there are not enough LSPs to comprise a significant fraction of the dark matter.