  We present a novel method for calculating the fundamental gap. To this end,
reduced-density-matrix-functional theory is generalized to fractional particle
number. For each fixed particle number, $M$, the total energy is minimized with
respect to the natural orbitals and their occupation numbers. This leads to a
function, $E_{\mathrm{tot}}^M$, whose derivative with respect to the particle
number has a discontinuity identical to the gap. In contrast to density
functional theory, the energy minimum is generally not a stationary point of
the total-energy functional. Numerical results, presented for alkali atoms, the
LiH molecule, the periodic one-dimensional LiH chain, and solid Ne, are in
excellent agreement with CI calculations and/or experimental data.
