  The expectation value of the complex phase factor of the fermion determinant
is computed to leading order in the $p$-expansion of the chiral Lagrangian. The
computation is valid for $\mu<m_\pi/2$ and determines the dependence of the
sign problem on the volume and on the geometric shape of the volume. In the
thermodynamic limit with $ L_i \to \infty $ at fixed temperature $1/L_0$, the
average phase factor vanishes. In the low temperature limit where $L_i/L_0$ is
fixed as $L_i$ becomes large the average phase factor approaches one. The
results for a finite volume compare well with lattice results obtained by
Allton {\it et al}.. After taking appropriate limits, we reproduce previously
derived results for the $\epsilon$-regime and for 1-dimensional QCD. The
distribution of the phase itself is also computed.
