  The relation between the on-shell and $\bar{\rm MS}$ 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.
