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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, $\dotω=0.138 \pm 0.002$ $\rm deg$ $\rm yr^{-1}$. If this induced $\dotω$ 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.