  The pulsar PSR J1756$-$2251 resides in a relativistic double neutron star
(DNS) binary system with a 7.67-hr orbit. We have conducted long-term precision
timing on more than 9 years of data acquired from five telescopes, measuring
five post-Keplerian parameters. This has led to several independent tests of
general relativity (GR), the most constraining of which shows agreement with
the prediction of GR at the 4% level. Our measurement of the orbital decay rate
disagrees with that predicted by GR, likely due to systematic observational
biases. We have derived the pulsar distance from parallax and orbital decay
measurements to be 0.73$_{-0.24}^{+0.60}$ kpc (68%) and < 1.2 kpc (95% upper
limit), respectively; these are significantly discrepant from the distance
estimated using Galactic electron density models. We have found the pulsar mass
to be 1.341$\pm$0.007 M$_\odot$, and a low neutron star (NS) companion mass of
1.230$\pm$0.007 M$_\odot$. We also determined an upper limit to the spin-orbit
misalignment angle of 34{\deg} (95%) based on a system geometry fit to
long-term profile width measurements. These and other observed properties have
led us to hypothesize an evolution involving a low mass loss, symmetric
supernova progenitor to the second-formed NS companion, as is thought to be the
case for the double pulsar system PSR J0737$-$3039A/B. This would make PSR
J1756$-$2251 the second compact binary system providing concrete evidence for
this type of NS formation channel.
