We prove an integral version of the Schur--Weyl duality between the
specialized Birman--Murakami--Wenzl algebra $B_n(-q^{2m+1},q)$ and the quantum
algebra associated to the symplectic Lie algebra sp_{2m}. In particular, we
deduce that this Schur--Weyl duality holds over arbitrary (commutative) ground
rings, which answers a question of Lehrer and Zhang [Strongly multiplicity free
modules for Lie algebras and quantum groups, J. Algebra (1) 306 (2006),
138--174] in the symplectic case. As a byproduct, we show that, as
$Z[q,q^{-1}]$-algebra, the quantized coordinate algebra defined by Kashiwara is
isomorphic to the quantized coordinate algebra arising from a generalized
Faddeev--Reshetikhin--Takhtajan's construction.