  Let B_n be the Artin braid group on n strings with standard generators
sigma_1, ..., sigma_{n-1}, and let SB_n be the singular braid monoid with
generators sigma_1^{+-1}, ..., sigma_{n-1}^{+-1}, tau_1, ..., tau_{n-1}. The
desingularization map is the multiplicative homomorphism eta: SB_n --> Z[B_n]
defined by eta(sigma_i^{+-1}) =_i^{+-1} and eta(tau_i) = sigma_i -
sigma_i^{-1}, for 1 <= i <= n-1. The purpose of the present paper is to prove
Birman's conjecture, namely, that the desingularization map eta is injective.
