  We give a new fusion procedure for the Brauer algebra by showing that all
primitive idempotents can be found by evaluating a rational function in several
variables which has the form of a product of R-matrix type factors. In
particular, this provides a new fusion procedure for the symmetric group
involving an arbitrary parameter. The R-matrices are solutions of the
Yang--Baxter equation associated with the classical Lie algebras g_N of types
B, C and D. Moreover, we construct an evaluation homomorphism from a reflection
equation algebra B(g_N) to U(g_N) and show that the fusion procedure provides
an equivalence between natural tensor representations of B(g_N) with the
corresponding evaluation modules.
