  Quantum corrections to the properties of a homogeneous interacting Bose gas
at zero temperature can be calculated as a low-density expansion in powers of
$\sqrt{\rho a^3}$, where $\rho$ is the number density and $a$ is the S-wave
scattering length. We calculate the ground state energy density to second order
in $\sqrt{\rho a^3}$. The coefficient of the $\rho a^3$ correction has a
logarithmic term that was calculated in 1959. We present the first calculation
of the constant under the logarithm. The constant depends not only on $a$, but
also on an extra parameter that describes the low energy $3\to 3$ scattering of
the bosons. In the case of alkali atoms, we argue that the second order quantum
correction is dominated by the logarithmic term, where the argument of the
logarithm is $\rho a \ell_V^2$, and $\ell_V$ is the length scale set by the van
der Waals potential.
