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.
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