  We report on a high-precision timing analysis and an astrophysical study of
the binary millisecond pulsar, PSR J1909$-$3744, motivated by the accumulation
of data with well improved quality over the past decade. Using 15 years of
observations with the Nan\c{c}ay Radio Telescope, we achieve a timing precision
of approximately 100 ns. We verify our timing results by using both broad-band
and sub-band template matching methods to create the pulse time-of-arrivals.
Compared with previous studies, we improve the measurement precision of secular
changes in orbital period and projected semi-major axis. We show that these
variations are both dominated by the relative motion between the pulsar system
and the solar system barycenter. Additionally, we identified four possible
solutions to the ascending node of the pulsar orbit, and measured a precise
kinetic distance of the system. Using our timing measurements and published
optical observations, we investigate the binary history of this system using
the stellar evolution code MESA, and discuss solutions based on detailed WD
cooling at the edge of the WD age dichotomy paradigm. We determine the 3-D
velocity of the system and show that it has been undergoing a highly eccentric
orbit around the centre of our Galaxy. Furthermore, we set up a constraint over
dipolar gravitational radiation with the system, which is complementary to
previous studies given the mass of the pulsar. We also obtain a new limit on
the parameterised post-Newtonian parameter, $\alpha_1<2.1 \times 10^{-5}$ at 95
% confidence level, which is fractionally better than previous best published
value and achieved with a more concrete method.
