The problem of recovering a measure \(\mu\) supported on a lattice of span \(\Delta\) from measurements \(\hat\mu(\omega)\) at frequencies \(|\omega|\leq\Omega\) is considered. The author shows that even if \(\Omega\) is much smaller than the Nyquist frequency \(\pi/\Delta\) and the measurements are noisy, stable recovery is possible if the measure \(\mu\) satisfies certain sparsity constraints. If the support of \(\mu\) is known a priori to have Rayleigh index at most \(R\) (in any interval of length \(4\pi/\Omega R\) there are at most \(R\) elements) then stable recovery is possible with a stability coefficient that grows at most like \(\Delta^{-2R-1}\) as \(\Delta\to 0\).