  For a translation invariant system of $N$ bosons in the Gross-Pitaevskii
regime, we establish a precise bound for the ground state energy $E_N$. While
the leading, order $N$, contribution to $E_N$ has been known since [30,28] and
the second order corrections (of order one) have been first determined in [5],
our estimate also resolves the next term in the asymptotic expansion of $E_N$,
which is of the order $(\log N) / N$.
