We prove the Paley-Wiener theorem for the spherical transform on the complex Grassmann manifolds
U
/
K
=
SU
(
p
+
q
)
/
S
(
U
p
×
U
q
)
U/K=\mbox {SU}(p+q)/\mbox {S}(\mbox {U}_p\times \mbox {U}_q)
. This theorem characterizes the
K
K
-biinvariant smooth functions
f
f
on the group
U
U
that are supported in the
K
K
-invariant ball of radius
R
R
, with
R
R
less than the injectivity radius of
U
/
K
U/K
, in terms of holomorphic extendability, exponential growth, and Weyl invariance properties of the spherical Fourier transforms
f
^
\hat {f}
, originally defined on the discrete set
Λ
s
p
h
\Lambda _{sph}
of highest restricted spherical weights.