  We give a realization of the stable L\'evy forest of a given size conditioned
by its mass from the path of the unconditioned forest. Then, we prove an
invariance principle for this conditioned forest by considering $k$ independent
Galton-Watson trees whose offspring distribution is in the domain of attraction
of any stable law conditioned on their total progeny to be equal to $n$. We
prove that when $n$ and $k$ tend towards $+\infty$, under suitable rescaling,
the associated coding random walk, the contour and height processes converge in
law on the Skorokhod space respectively towards the "first passage bridge" of a
stable L\'evy process with no negative jumps and its height process.
