We show that the gravitational binding energy of a neutron star of a given mass is correlated with the slope of the nuclear symmetry energy at 1-2 times nuclear saturation density for equations of state without significant softening ( i.e. , those that predict maximum masses M _ { max } > 1.44 M _ { \odot } in line with the largest accurately measured neutron star mass ) .
Applying recent laboratory constraints on the slope of the symmetry energy to this correlation we extract a constraint on the baryon mass of the lower mass member of the double pulsar binary system , PSR J0737-3039B .
We compare with independent constraints derived from modeling the progenitor star of J0737-3039B up to and through its collapse under the assumption that it formed in an electron capture supernova .
The two sets of constraints are consistent only if L \lesssim 70 MeV .