The relation between the on-shell and $\overline{\rm MS}$ mass can be
expressed through scalar and vector part of the quark propagator. In
principle these two-point functions have to be evaluated on-shell
which is a non-trivial task at three-loop order. Instead, we evaluate
the quark self energy in the limit of large and small external
momentum and use conformal mapping in combination with Pad\'e
improvement in order to construct a numerical approximation for the
relation [1].
The errors of our final result are conservatively estimated
to be below 3\%. The numerical implications of the results are
discussed in particular in view of top and bottom quark production
near threshold. We show that the knowledge of new ${\cal
O}(\alpha_s^3)$ correction leads to a significant reduction of the
theoretical uncertainty in the determination of the quark masses.
