A form of the Peierls free-energy variational theorem is applied to the Heisenberg Hamiltonian for a three-dimensional system with nearest-neighbor antiferromagnetic interaction. For a large magnetic field (h≡gμH4SJz~1) we find a phase boundary separating a region of antiferromagnetic order from one of ferromagnetic order. At low temperatures (θ≡kT2SJz<<1) the phase boundary has the leading behavior: h=1-aθ32 with a=2ζ (32) (32π)32S for a simple cubic antiferromagnetic lattice (e.g., RbMnF3). At the phase boundary the magnetization is continuous; whereas a discontinuity in the susceptibility is suggested but not firmly established by this treatment. Low-temperature expressions are given for the magnetization, susceptibility, and specific heat above the boundary. Numerical calculations show that, for the approximation used, the phase boundary extends to a maximum θ at which the magnetization is nonzero. For the limiting case of h=0 we obtain Keffer and Loudon's renormalized spectrum and magnetization for a ferromagnet and for an antiferromagnet from a single variational calculation. Attention is also given to a reduced Hamiltonian which, when treated by the variational method, exhibits the properties of an antiferromagnetic molecular field model previously proposed by Garrett for S=12.