  We consider N bosons in a box with volume one, interacting through a two-body
potential with scattering length of the order $N^{-1+\kappa}$, for $\kappa>0$.
Assuming that $\kappa\in (0;1/43)$, we show that low-energy states of the
system exhibit complete Bose-Einstein condensation by providing explicit bounds
on the expectation and on higher moments of the number of excitations.
