  According to a general probabilistic principle, the natural divisors of
friable integers (i.e.~free of large prime factors) should normally present a
Gaussian distribution. We show that this indeed is the case with conditional
density tending to 1 as soon as the standard necessary conditions are met.
Furthermore, we provide explicit, essentially optimal estimates for the decay
of the involved error terms. The size of the exceptional set is sufficiently
small to enable recovery of the average behaviour in the same optimal range.
Our argument combines the saddle-point method with new large deviations
estimates for the distribution of certain additive functions.
