  Precise determinations of the masses of the $W$ boson and of the top quark
could stringently test the radiative structure of the Standard Model (SM) or
provide evidence for new physics. We analyze the excellent prospects at a muon
collider for measuring $M_W$ and $m_t$ in the $W^+W^-$ and $t\bar t$ threshold
regions. With an integrated luminosity of 10 (100) fb$^{-1}$, the $W$-boson
mass could be measured to a precision of 20 (6) MeV, and the top-quark mass to
a precision of 200 (70) MeV, provided that theoretical and experimental
systematics are understood. A measurement of $\Delta m_t=200$ MeV for fixed
$M_W$ would constrain a 100 GeV SM Higgs mass within about $\pm 2$ GeV, while
$\Delta M_W=6$ MeV for fixed $m_t$ would constrain $m_h$ to about $\pm 10$ GeV.
