  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 $M_{\rm max} > 1.44M_{\odot}$ 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.
