  The minimal integral Mahler measure of a number field $K$,
$M(\mathcal{O}_K)$, is the minimal Mahler measure of a non-torsion primitive
element of $\mathcal{O}_K$. Upper and lower bounds, which depend on the
discriminant, are known. We show that for cubics, the lower bounds are sharp
with respect to its growth as a function of discriminant. We construct an
algorithm to compute $M(\mathcal{O}_K)$ for all cubics with absolute value of
the discriminant bounded by $N$.
