The Tikhonov functional with the $\ell^1$ penalty yields a regularization
method that generates a sparse approximate solution--the so-called Tikhonov
regularization with sparsity constraints. Recently, it has been shown that this
functional together with a certain a priori parameter rule and a certain source
condition converges linearly to the minimum-$\ell^1$ solution. In this paper we
go beyond the question of convergence rates by presenting an a priori parameter
rule which ensures exact recovery of the unknown support.