  Given any postsingularly finite exponential function $p_\lambda(z) = \lambda
\exp(z)$ where $\lambda \in \C^*$, we construct a sequence of postcritically
finite unicritical polynomials $p_{d,\lambda_d}(z) =
\lambda_d(1+\frac{z}{d})^d$ that converge to $p_\lambda$ locally uniformly in
$\C$, with the same postsingular portrait as that of $p_\lambda$. We describe
$\lambda_d$ in terms of parameter rays in the space of degree $d$ unicritical
polynomials, and exhibit a relationship between the angles of these parameter
rays as $d \rightarrow \infty$ and the external addresses associated with
$\lambda$ in the exponential parameter plane.
