  We investigate the age constraints that can be placed on the double pulsar
system using models for the spin-down of the first-born 22.7-ms pulsar A and
the 2.77-s pulsar B with characteristic ages of 210 and 50 Myr respectively.
Standard models assuming dipolar spin-down of both pulsars suggest that the
time since the formation of B is ~50 Myr, i.e. close to B's characteristic age.
However, adopting models which account for the impact of A's relativistic wind
on B's spin-down we find that the formation of B took place either 80 or 180
Myr ago, depending the interaction mechanism. Formation 80 Myr ago, closer to
B's characteristic age, would result in the contribution from J0737-3039 to the
inferred coalescence rates for double neutron star binaries increasing by 40%.
The 180 Myr age is closer to A's characteristic age and would be consistent
with the most recent estimates of the coalescence rate. The new age constraints
do not significantly impact recent estimates of the kick velocity, tilt angle
between pre and post-supernova orbital planes or pre-supernova mass of B's
progenitor.
