  We derive a upper bound on the free energy of a Bose gas system at density
$\rho$ and temperature $T$. In combination with the lower bound derived
previously by Seiringer \cite{RS1}, our result proves that in the low density
limit, i.e., when $a^3\rho\ll 1$, where $a$ denotes the scattering length of
the pair-interaction potential, the leading term of $\Delta f$ the free energy
difference per volume between interacting and ideal Bose gases is equal to
$4\pi a (2\rho^2-[\rho-\rhoc]^2_+)$. Here, $\rhoc(T)$ denotes the critical
density for Bose-Einstein condensation (for the ideal gas), and $[\cdot ]_+$
$=$ $\max\{\cdot, 0\}$ denotes the positive part.
