  We compute analytically the complete electroweak two-loop corrections to the
$B^0-{\bar B}^0$ mixing. These corrections fix the normalization of the
electroweak coupling employed in the extraction of $|V_{td}|$ and reduce the
theoretical uncertainty due to higher order electroweak effects from several
percent to a few parts in a thousand. If the LO result is expressed in terms of
$G_\mu$ or of the $\bar{MS}$ coupling $\hat{g}(M_Z)$, the two-loop corrections
are $O(1%)$, the exact value depending on the mass of the Higgs boson. We
discuss in detail the renormalization procedure and the scheme and scale
dependence, and provide practical formulas for the numerical implementation of
our results. We also consider the heavy top mass expansion and show that in the
case at hand it converges very slowly.
