We study a dilute Bose gas of atoms whose scattering length [Formula presented] is large compared to the range of their interaction. We calculate the energy density [Formula presented] of a homogeneous Bose-Einstein condensate (BEC) to second order in the low-density expansion, expressing it in terms of [Formula presented] and a second parameter [Formula presented] that determines the low-energy observables in the three-body sector. The second-order correction to [Formula presented] has a small imaginary part that reflects the instability due to three-body recombination. In the case of a trapped BEC with large negative [Formula presented], we calculate the coefficient of the three-body mean-field term in [Formula presented] in terms of [Formula presented] and [Formula presented]. It can be very large if there is an Efimov state near threshold. © 2002 The American Physical Society.