The minimal supersymmetric standard model (MSSM) predicts the existence of a massive stable particle (the lightest supersymmetric particle, or LSP) which could make up the dark matter in the universe. The LSP is commonly considered to be either a photino or a certain linear combination of higgsinos. However, when the supersymmetry-breaking gaugino and higgsino masses are taken to be large (but ≲ 10 TeV), the LSP is typically either a bino (superpartner of the hypercharge gauge boson) or a different linear combination of higgsinos. We calculate in detail the annihilation cross section for these massive (m ≳ 20 GeV)LSP's, and use the results to determine their relic abundance. We show that if the LSP is a bino, both it and at least one squark or slepton must have mass less than ∼ 350 GeV in order to avoid overclosing the universe. If the LSP is a higgsino, its mass must be less than ∼ 1 TeV in order to avoid overclosing the universe. We also study in detail the prospect that the dark matter is composed of one of these particles.