The Double Pulsar (PSR J0737-3039) is the only neutron star-neutron star (NS-NS) binary in which both NSs have been detectable as radio pulsars. The Double Pulsar has been assumed to dominate the Galactic NS-NS binary merger rate R_g among all known systems, solely based on the properties of the first-born, recycled pulsar (PSR J0737-3039A, or A) with an assumption for the beaming correction factor of 6. In this work, we carefully correct observational biases for the second-born, non-recycled pulsar (PSR J0737-0737B, or B) and estimate the contribution from the Double Pulsar on R_g using constraints available from both A and B. Observational constraints from the B pulsar favour a small beaming correction factor for A (~2), which is consistent with a bipolar model. Considering known NS-NS binaries with the best observational constraints, including both A and B, we obtain R_g=21_{-14}^{+28} per Myr at 95 per cent confidence from our reference model. We expect the detection rate of gravitational waves from NS-NS inspirals for the advanced ground-based gravitational-wave detectors is to be 8^{+10}_{-5} per yr at 95 per cent confidence. Within several years, gravitational-wave detections relevant to NS-NS inspirals will provide us useful information to improve pulsar population models.
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