  In this paper we compute the three-loop corrections to the $\beta$ function
in a momentum subtraction (MOM) scheme with a massive quark. The calculation is
performed in the background field formalism applying asymptotic expansions for
small and large momenta. Special emphasis is devoted to the relation between
the coupling constant in the MOM and $\overline{\rm{MS}}$ schemes as well as
their ability to describe the phenomenon of decoupling.
  It is demonstrated by an explicit comparison that the $\overline{\rm{MS}}$
scheme can be consistently used to relate the values of the MOM-scheme
strong-coupling constant in the energy regions higher and lower than the
massive-quark production threshold. This procedure obviates the necessity to
know the full mass dependence of the MOM $\beta$ function and clearly
demonstrates the equivalence of both schemes for the description of physics
outside the threshold region.
