  We consider an interacting, dilute Bose gas trapped in a harmonic potential
at a positive temperature. The system is analyzed in a combination of a
thermodynamic and a Gross-Pitaevskii (GP) limit where the trap frequency
$\omega$, the temperature $T$ and the particle number $N$ are related by $N
\sim (T / \omega)^{3} \to\infty$ while the scattering length is so small that
the interaction energy per particle around the center of the trap is of the
same order of magnitude as the spectral gap in the trap. We prove that the
difference between the canonical free energy of the interacting gas and the one
of the noninteracting system can be obtained by minimizing the GP energy
functional. We also prove Bose-Einstein condensation in the following sense:
The one-particle density matrix of any approximate minimizer of the canonical
free energy functional is to leading order given by that of the noninteracting
gas but with the free condensate wavefunction replaced by the GP minimizer.
