Let {ξ
k
: k ≧ 1} be a sequence of independent, identically distributed random variables with E{ξ
1} = μ ≠ 0. Form the random walk {S
n
: n ≧ 0} by setting S
0, S
n
= ξ
1 + ξ
2 + ··· + ξ
n
, n ≧ 1. Define the random function Xn
by setting where α is a norming constant. Let N denote the hitting time of the set (–∞, 0] by the random walk. The principal result in this paper is to show (under appropriate conditions on the distribution of ξ
1) that the finite-dimensional distributions of Xn
, conditioned on n < N < ∞ converge to those of the Brownian excursion process.