  We analyze the suppression of the phase stiffness in a superconductor by
antiferromagnetic order. The analysis is based on a general expression for the
phase stiffness in a mean-field state with coexisting spin-singlet
superconductivity and spiral magnetism. Neel order is included as a special
case. Close to half-filling, where the pairing gap is much smaller than the
magnetic gap, a simple formula for the phase stiffness in terms of magnetic
quasi-particle bands is derived. The phase stiffness is determined by charge
carriers in small electron or hole pockets in this regime. The general analysis
is complemented by a numerical calculation for the two-dimensional Hubbard
model with nearest and next-to-nearest neighbor hopping amplitudes at a
moderate interaction strength. The resulting phase stiffness exhibits a
striking electron-hole asymmetry. In the ground state, it is larger than the
pairing gap on the hole-doped side, and smaller for electron doping. Hence, in
the hole-doped regime near half-filling the ground state pairing gap sets the
scale for the Kosterlitz-Thouless temperature T_c^{KT}, while in the slightly
electron-doped regime T_c^{KT} is determined essentially by the ground state
phase stiffness.
