  Following a suggestion of Kostelecky et al. we have evaluated a test of CPT
and Lorentz invariance from the microwave spectroscopy of muonium. Hamiltonian
terms beyond the standard model violating CPT and Lorentz invariance would
contribute frequency shifts $\delta\nu_{12}$ and $\delta\nu_{34}$ to $\nu_{12}$
and $\nu_{34}$, the two transitions involving muon spin flip, which were
precisely measured in ground state muonium in a strong magnetic field of 1.7 T.
The shifts would be indicated by anti-correlated oscillations in $\nu_{12}$ and
$\nu_{34}$ at the earth's sidereal frequency. No time dependence was found in
$\nu_{12}$ or $\nu_{34}$ at the level of 20 Hz, limiting the size of some CPT
and Lorentz violating parameters at the level of $2\times10^{-23}$ GeV,
representing Planck scale sensitivity and an order of magnitude improvement in
sensitivity over previous limits for the muon.
