To understand the nature of supernovae and neutron star (NS) formation, as well as binary stellar evolution and their interactions, it is important to probe the distribution of NS masses. Until now, all double NS (DNS) systems have been measured as having a mass ratio close to unity (q≥ 0.91). Here, we report the measurement of the individual masses of the 4.07-day binary pulsar J0453+ 1559 from measurements of the rate of advance of periastron and Shapiro delay: the mass of the pulsar is M p= 1.559±0.005 M⊙ and that of its companion is M⊙; q= 0.75. If this companion is also an NS, as indicated by the orbital eccentricity of the system (e= 0.11), then its mass is the smallest precisely measured for any such object. The pulsar has a spin period of 45.7 ms and a spin period derivative of =(1.8616±0.0007)× 10− 19 ss− 1; from these, we derive a characteristic age of∼ 4.1× 10 9 years and a magnetic field of∼ 2.9× 10 9 G, ie, this pulsar was mildly recycled by the accretion of matter from the progenitor of the companion star. This suggests that it was formed with (very approximately) its current mass. Thus, NSs form with a wide range of masses, which is important for understanding their formation in supernovae. It is also important for the search for gravitational waves released during an NS–NS merger: it is now evident that we should not assume that all DNS systems are symmetric.