We prove an integral version of the Schur–Weyl duality between the specialized Birman–Murakami–Wenzl algebra
B
n
(
−
q
2
m
+
1
,
q
)
\mathfrak {B}_n(-q^{2m+1},q)
and the quantum algebra associated to the symplectic Lie algebra
s
p
2
m
\mathfrak {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 in the symplectic case. As a byproduct, we show that, as a
Z
[
q
,
q
−
1
]
\mathbb {Z}[q,q^{-1}]
-algebra, the quantized coordinate algebra defined by Kashiwara (which he denoted by
A
q
Z
(
g
)
A_q^{\mathbb {Z}}(g)
) is isomorphic to the quantized coordinate algebra arising from a generalized Faddeev–Reshetikhin–Takhtajan construction.