Recent work has shown that convex programming allows to recover a superposition of point sources exactly from low-resolution data as long as the sources are separated by 2/fc, where fc is the cut-off frequency of the sensing process. The proof relies on the construction of a certificate whose existence implies exact recovery. This certificate has since been used to establish that the approach is robust to noise and to analyze related problems such as compressed sensing off the grid and the super-resolution of splines from moment measurements. In this work we construct a new certificate that allows to extend all these results to signals with minimum separations above 1.26/fc. This is close to 1/fc, the threshold at which the problem becomes inherently ill posed, in the sense that signals with a smaller minimum separation may have low-pass projections with negligible energy.