  The double pulsar J0737--3039 has become an important astrophysical
laboratory for testing fundamental physics. Here we demonstrate that the low
measured mass of Pulsar B can be used to constrain the equation of state of
neutron star matter {\em under the assumption} that it formed in an
electron-capture supernova. We show that the observed orbital parameters as
well as the likely evolutionary history of the system support such a hypothesis
and discuss future refinements that will improve the constraints this test may
provide.
