Ultraluminous x-ray sources (ULXs) in nearby galaxies shine brighter than any X-ray source in our Galaxy. ULXs are usually modeled as stellar-mass black holes (BHs) accreting at very high rates or intermediate-mass BHs. We present observations showing that NGC5907 ULX is instead an x-ray accreting neutron star (NS) with a spin period evolving from 1.43~s in 2003 to 1.13~s in 2014. It has an isotropic peak luminosity of about 1000 times the Eddington limit for a NS at 17.1~Mpc. Standard accretion models fail to explain its luminosity, even assuming beamed emission, but a strong multipolar magnetic field can describe its properties. These findings suggest that other extreme ULXs (x-ray luminosity > 10^{41} erg/s) might harbor NSs.
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