We consider the quantum Heisenberg antiferromagnet on a face-centered-cubic lattice in which $J$, the second-neighbor (intrasublattice) exchange constant, dominates ${J}^{\ensuremath{'}}$, the first-neighbor (intersublattice) exchange constant. It is shown that the continuous degeneracy of the classical ground state with four decoupled (in a mean-field sense) simple cubic antiferromagnetic sublattices is removed so that at second order in ${J}^{\ensuremath{'}}/J$ the spins are collinear. Here we study the degeneracy between the two inequivalent collinear structures by analyzing the contribution to the spin-wave zero-point energy which is of the form ${\mathcal{H}}_{\mathrm{eff}}{/J=C}_{0}{+C}_{4}{\ensuremath{\sigma}}_{1}{\ensuremath{\sigma}}_{2}{\ensuremath{\sigma}}_{3}{\ensuremath{\sigma}}_{4}{(J}^{\ensuremath{'}}{/J)}^{4}{+O(J}^{\ensuremath{'}}{/J)}^{5},$ where ${\ensuremath{\sigma}}_{i}$ specifies the phase of the $i$th collinear sublattice, ${C}_{0}$ depends on ${J}^{\ensuremath{'}}/J$ but not on the $\ensuremath{\sigma}$'s, and ${C}_{4}$ is a positive constant. Thus the ground state is one in which the product of the $\ensuremath{\sigma}$'s is $\ensuremath{-}1.$ This state, known as the second kind of type A, is stable in the range $|{J}^{\ensuremath{'}}|<2|J|$ for large $S.$ Using interacting spin-wave theory, it is shown that the main effect of the zero-point fluctuations is at small wave vector and can be well modeled by an effective biquadratic interaction of the form $\ensuremath{\Delta}{E}_{Q}^{\mathrm{eff}}=\ensuremath{-}\frac{1}{2}Q{\ensuremath{\sum}}_{i,j}[\mathbf{S}(i)\ensuremath{\cdot}\mathbf{S}(j){]}^{2}{/S}^{3}.$ This interaction opens a spin gap by causing the extra classical zero-energy modes to have a nonzero energy of order ${J}^{\ensuremath{'}}\sqrt{S}.$ We also study the dependence of the zero-point spin reduction on ${J}^{\ensuremath{'}}/J$ and the sublattice magnetization on temperature. The resulting experimental consequences are discussed.