  Ultraluminous X-ray sources (ULX) are off-nuclear point sources in nearby
galaxies whose X-ray luminosity exceeds the theoretical maximum for spherical
infall (the Eddington limit) onto stellar-mass black holes. Their luminosity
ranges from $10^{40}$ erg s$^{-1} < L_X$(0.5 - 10 keV) $<10^{40}$ erg s$^{-1}$.
Since higher masses imply less extreme ratios of the luminosity to the
isotropic Eddington limit theoretical models have focused on black hole rather
than neutron star systems. The most challenging sources to explain are those at
the luminous end ($L_X$ > $10^{40}$ erg s$^{-1}$), which require black hole
masses MBH >50 solar masses and/or significant departures from the standard
thin disk accretion that powers bright Galactic X-ray binaries. Here we report
broadband X-ray observations of the nuclear region of the galaxy M82, which
contains two bright ULXs. The observations reveal pulsations of average period
1.37 s with a 2.5-day sinusoidal modulation. The pulsations result from the
rotation of a magnetized neutron star, and the modulation arises from its
binary orbit. The pulsed flux alone corresponds to $L_X$(3 - 30 keV) = $4.9
\times 10^{39}$ erg s$^{-1}$. The pulsating source is spatially coincident with
a variable ULX which can reach $L_X$ (0.3 - 10 keV) = $1.8 \times 10^{40}$ erg
s$^{-1}$. This association implies a luminosity ~100 times the Eddington limit
for a 1.4 solar mass object, or more than ten times brighter than any known
accreting pulsar. This finding implies that neutron stars may not be rare in
the ULX population, and it challenges physical models for the accretion of
matter onto magnetized compact objects.
