We study the two-dimensional process of integrated Brownian motion and Brow
nian motion, where integrated Brownian motion is conditioned to be positive
. The transition density of this process is derived from the asymptotic beh
avior of hitting times of the unconditioned process. Explicit expressions f
or the transition density in terms of confluent hypergeometric functions ar
e derived, and it is shown how our results on the hitting time distribution
s imply previous results of Isozaki-Watanabe and Goldman, The conditioned p
rocess is characterized by a system of stochastic differential equations (S
DEs) for which we prove an existence and unicity result. Some sample path p
roperties are derived from the SDEs and it is shown that t --> t(9/10) is a
"critical curve" for the conditioned process in the sense that the expecte
d time that the integral part of the conditioned process spends below any c
urve t --> t(alpha) is finite for alpha < 9/10 and infinite for alpha great
er than or equal to 9/10.