We present a field theoretical equation of state for nuclear matter and neutron-rich matter in β equilibrium using the chiral sigma model. The model includes an isoscalar vector field generated dynamically and reproduces the empirical values of the nuclear matter saturation density and binding energy and the iso spin symmetry coefficient for asymmetric nuclear matter. The energy per nucleon of nuclear matter as predicted by our calculation is in very good agreement, up to about a density of 4ns (ns = nucleon number density for saturating nuclear matter), with the estimates inferred from heavy-ion collision data. An astrophysical application, relating to neutron star structure, is presented. An extension of the nuclear matter equation of state for finite temperatures (up to 15 MeV) is given, and the phase transition to (u, d, s) quark matter is discussed.