  In 240 days of X-ray monitoring of M82, we have discovered an X-ray
periodicity at $62.0 \pm 2.5$ days with a peak to peak amplitude corresponding
to an isotropic luminosity of $2.4 \times 10^{40} \rm erg s^{-1}$ in M82 and an
X-ray flare reaching a peak luminosity of $9.8 \times 10^{40} \rm erg s^{-1}$.
The periodicity and flare likely originate from the ultraluminous X-ray source
(ULX) in M82 which has been identified as a possible intermediate mass black
hole. We suggest that the 62~day modulation is due to orbital motion within an
X-ray binary with a Roche-lobe overflowing companion star which would imply
that the average density of the companion star is near $5 \times 10^{-5} \rm g
cm^{-3}$ and is therefore a giant or supergiant. Chandra observations just
after the flare show an energy spectrum that is consistent with a power-law
with no evidence of a thermal component or line emission. Radio observations
made with the VLA during the flare allow us to rule out a blazar identification
for the source and place strong constraints on relativistically beamed models
of the X-ray emission. The Chandra observations reveal a second X-ray source
reached a flux of $4.4 \times 10^{-12} \rm erg cm^{-2} s^{-1}$ in the 0.3-7 keV
band which is dramatically higher than any flux previously seen from this
source and corresponds to an isotropic luminosity of $1.1 \times 10^{40} \rm
erg s^{-1}$. This source is a second ultraluminous X-ray source in M82 and may
give rise to the QPOs detected from the central region of M82.
