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Let $M$ be a closed oriented surface of genus $g\ge 1$, let $B_n(M)$ be thebraid group of $M$ on $n$ strings, and let $SB_n(M)$ be the correspondingsingular braid monoid. Our purpose in this paper is to prove that thedesingularization map $\eta: SB_n(M) \to \Z [B_n(M)]$, introduced in thedefinition of the Vassiliev invariants (for braids on surfaces), is injective.

BIRMAN'S CONJECTURE FOR SINGULAR BRAIDS ON CLOSED SURFACES