The fundamental properties of neutron stars provide a direct test of the equation of state of cold nuclear matter, a relationship between pressure and density that is determined by the physics of the strong interactions between the particles that constitute the star. The most straightforward method of determining these properties is by measuring the gravitational redshift of spectral lines produced in the neutron star photosphere. The equation of state implies a mass-radius relation, while a measurement of the gravitational redshift at the surface of a neutron star provides a direct constraint on the mass-to-radius ratio. Here we report the discovery of significant absorption lines in the spectra of 28 bursts of the low-mass X-ray binary EXO0748-676. We identify the most significant features with the Fe XXVI and XXV n = 2-3 and O VIII n = 1-2 transitions, all with a redshift of z = 0.35, identical within small uncertainties for the respective transitions. For an astrophysically plausible range of masses (M ~ 1.3-2.0 solar masses; refs 2-5), this value is completely consistent with models of neutron stars composed of normal nuclear matter, while it excludes some models in which the neutron stars are made of more exotic matter.