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.
