  We study the relationship between the $\overline{\mathrm{MS}}$ Yukawa
coupling and the pole mass for the bottom and top quarks at the two-loop
electroweak order ${\cal O}(\alpha^2)$ in the gaugeless limit of the standard
model. We also consider the $\overline{\mathrm{MS}}$ to pole mass relationships
at this order, which include tadpole contributions to ensure the gauge
independence of the $\overline{\mathrm{MS}}$ masses. In order to suppress
numerically large tadpole contributions, we propose a redefinition of the
running heavy-quark mass in terms of the $\overline{\mathrm{MS}}$ Yukawa
coupling. We also present $\Delta r$ in the $\overline{\mathrm{MS}}$ scheme at
${\cal O}(\alpha^2)$ in the gaugeless limit. As an aside, we also list the
exact two-loop expression for the mass counterterms of the bottom and top
quarks.
