  In this paper we provide a solution to the double coset problem for the braid
group $B_n$ modulo the Hilden subgroup $H_n.$ This result demonstrates that, as
in the case of braid closures, the Link Problem for plat closures is "stably
equivalent" to a solvable algebraic problem. A particularly interesting feature
of the proof is that, like Garside's solutions to the Word and Conjugacy
Problems, it too relies on Garside's decomposition of braids in $B_n.$
