This paper utilizes the second-order Green's-function formalism to treat a Heisenberg ferromagnet in the presence of single-ion crystal-field anisotropy represented by the Hamiltonian: H=-Ji, (Si Si +)-Di(Siz)2. Second-order equations of motion for the Green's functions Siz; Sjz and Si+; Sj- are developed and the higher-order Green's functions are decoupled in the zeroth-order approximation in which the interspin correlations are not taken into account rigorously. The spin-correlation functions are derived and are solved self-consistently in the limit of zero spontaneous magnetization. The Curie temperature is thus obtained. Calculations are restricted to a simple-cubic lattice and positive values of D only, and the sensitivity of the Curie temperature to the single-ion anisotropy is critically examined for a spin-1 lattice. It is seen that the results agree very closely with those of the Green's-function diagram technique. The results are also compared with those of the first-order Green's-function theory using the random-phase approximations of Lines using the correlated-effective-field approximation, and of the molecular-field theory. © 1980 The American Physical Society.