  We study sparse spikes deconvolution over the space of Radon measures on
$\mathbb{R}$ or $\mathbb{T}$ when the input measure is a finite sum of positive
Dirac masses using the BLASSO convex program. We focus on the recovery
properties of the support and the amplitudes of the initial measure in the
presence of noise as a function of the minimum separation $t$ of the input
measure (the minimum distance between two spikes). We show that when
${w}/\lambda$, ${w}/t^{2N-1}$ and $\lambda/t^{2N-1}$ are small enough (where
$\lambda$ is the regularization parameter, $w$ the noise and $N$ the number of
spikes), which corresponds roughly to a sufficient signal-to-noise ratio and a
noise level small enough with respect to the minimum separation, there exists a
unique solution to the BLASSO program with exactly the same number of spikes as
the original measure. We show that the amplitudes and positions of the spikes
of the solution both converge toward those of the input measure when the noise
and the regularization parameter drops to zero faster than $t^{2N-1}$.
