  We compute the persistence exponent of the integral of a stable L\'evy
process in terms of its self-similarity and positivity parameters. This solves
a problem raised by Z. Shi (2003). Along the way, we investigate the law of the
stable process L evaluated at the first time its integral X hits zero, when the
bivariate process (X,L) starts from a coordinate axis. This extends classical
formulae by McKean (1963) and Gor'kov (1975) for integrated Brownian motion.
