We report on variational calculations of the energy E( ρ, β) of asymmetric nuclear matter having ϱ = ϱ n + ϱ p = 0.05 to 0.35 fm −3, and β = ( ϱ n − ϱ p / g9 = 0 to 1. The nuclear h used in this work consists of a realistic two-nucleon interaction, called v 14, that fits the available nucleon-nucleon scattering data up to 425 MeV, and a phenomenological three nucleon interaction adjusted to reproduce the empirical properties of symmetric nuclear matter. The variational many-body theory of symmetric nuclear matter is extended to treat matter with neutron excess. Numerical and analytic studies of the β-dependence of various contributions to the nuclear matter energy show that at ϱ < 0.35 fm −3 the β 4 terms are very small, and that the interaction energy EI(ρ, β) defined as E( ρ, β) − T F( ρ, β), where T F is the Fermi-gas energy, is well approximated by EI 0(ϱ) + β 2EI 2(ρ). The calculated symmetry energy at equilibrium density is 30 MeV and it increases from 15 to 38 MeV as ϱ increases from 0.05 to 0.35 fm −3.