  We prove an existence and uniqueness result for the infinitely delayed
stochastic evolution equation $$dU(t) = &\big(AU(t) + F(t,U_t)\big) dt +
B(t,U_t)dW_H(t), t\in[0,T_0]$$ where $A$ is the generator of an analytic
semigroup on a UMD space $E$, $F$ and $B$ satisfy Lipschitz conditions and
$\mathscr{B}$ is a weighted $L^p$ history space. This paper is based on recent
work of van Neerven \emph{et al.}~which developed the theory of abstract
stochastic evolution equations in UMD spaces.
