  The observable population of double neutron star (DNS) systems in the Milky
Way allow us to understand the nature of supernovae and binary stellar
evolution. Until now, all DNS systems in wide orbits ($ P_{\textrm{orb}}>$
1~day) have been found to have orbital eccentricities, $e > 0.1$. In this
paper, we report the discovery of pulsar PSR J1325$-$6253: a DNS system in a
1.81 day orbit with a surprisingly low eccentricity of just $e = 0.064$.
Through 1.4 yr of dedicated timing with the Parkes radio telescope we have been
able to measure its rate of advance of periastron, $\dot{\omega}=0.138 \pm
0.002$ $\rm deg$ $\rm yr^{-1}$. If this induced $\dot{\omega}$ is solely due to
general relativity then the total mass of the system is, $M_{\rm sys} = 2.57
\pm 0.06$ M$_{\odot}$. Assuming an edge-on orbit the minimum companion mass is
constrained to be $M_\mathrm{c,min}>0.98$ M$_{\odot}$ which implies the pulsar
mass is $M_\mathrm{p,max}<1.59 $ M$_{\odot}$. Its location in the $P$-$\dot{P}$
diagram suggests that, like other DNS systems, PSR J1325$-$6253 is a recycled
pulsar and if its mass is similar to the known examples ($>1.3$ M$_\odot$),
then the companion neutron star is probably less than $\sim1.25$ M$_\odot$ and
the system is inclined at about $50^{\circ}$-$60^{\circ}$. The low eccentricity
along with the wide orbit of the system strongly favours a formation scenario
involving an ultra-stripped supernova explosion.
