  The Laplace transform of the first passage time density of the
Ornstein--Uhlenbeck process for a constant threshold contains a ratio of two
parabolic cylinder functions for which no analytical inversion formula is
available. Recently derived inverse Laplace transforms for the product of two
parabolic cylinder functions together with the convolution theorem of the
Laplace transform then allow to derive a new Volterra integral equation for
this first passage time density. The kernel of this integral equation contains
a parabolic cylinder function and the Fortet renewal equation for the
Ornstein-Uhlenbeck process emerges as a special case, namely when the order q
of the parabolic cylinder function is set at 0. The integral equation is shown
to hold both for constant as well as time dependent thresholds. Moreover, the
kernel of the integral equation is regular for q<=-1.
