  Calculation of statistical properties of nuclei in a finite-temperature
mean-field theory requires projection onto good particle number, since the
theory is formulated in the grand canonical ensemble. This projection is
usually carried out in a saddle-point approximation. Here we derive formulas
for an exact particle-number projection of the finite-temperature mean-field
solution. We consider both deformed nuclei, in which the pairing condensate is
weak and the Hartree-Fock (HF) approximation is the appropriate mean-field
theory, and nuclei with strong pairing condensates, in which the appropriate
theory is the Hartree-Fock-Bogoliubov (HFB) approximation, a method that
explicitly violates particle-number conservation. For the HFB approximation, we
present a general projection formula for a condensate that is time-reversal
invariant and a simpler formula for the Bardeen-Cooper-Schrieffer (BCS) limit,
which is realized in nuclei with spherical condensates. We apply the method to
three heavy nuclei: a typical deformed nucleus $^{162}$Dy, a typical spherical
nucleus $^{148}$Sm, and a transitional nucleus $^{150}$Sm in which the pairing
condensate is deformed. We compare the results of this projection with results
from the saddle-point approximation and exact shell model Monte Carlo
calculations. We find that the approximate canonical HF entropy in the
particle-number projection decreases monotonically to zero in the limit when
the temperature goes to zero. However, in a nucleus with a strong pairing
condensate, the approximate canonical HFB entropy in the particle-number
projection decreases monotonically to a negative value, reflecting the
violation of particle-number conservation. Computationally, the exact
particle-number projection is more efficient than calculating the derivatives
required in the saddle-point approximation.
