  Mergers of double neutron stars are considered the most likely progenitors
for short gamma-ray bursts. Indeed such a merger can produce a black hole with
a transient accreting torus of nuclear matter (Lee & Ramirez-Ruiz 2007,
Oechslin & Janka 2006), and the conversion of a fraction of the torus
mass-energy to radiation can power a gamma-ray burst (Nakar 2006). Using
available binary pulsar observations supported by our extensive evolutionary
calculations of double neutron star formation, we demonstrate that the fraction
of mergers that can form a black hole -- torus system depends very sensitively
on the (largely unknown) maximum neutron star mass. We show that the available
observations and models put a very stringent constraint on this maximum mass
under the assumption that a black hole formation is required to produce a short
gamma-ray burst in a double neutron star merger. Specifically, we find that the
maximum neutron star mass must be within 2 - 2.5 Msun. Moreover, a single
unambiguous measurement of a neutron star mass above 2.5 Msun would exclude a
black hole -- torus central engine model of short gamma-ray bursts in double
neutron star mergers. Such an observation would also indicate that if in fact
short gamma-ray bursts are connected to neutron star mergers, the gamma-ray
burst engine is best explained by the lesser known model invoking a highly
magnetized massive neutron star (e.g., Usov 1992; Kluzniak & Ruderman 1998; Dai
et al. 2006; Metzger, Quataert & Thompson 2007).
