  We investigate the realizations of a random Gaussian field on a finite domain
of ${\mathbb R}^d$ in the limit where a given linear functional of the field is
large. We prove that if its variance is bounded, the field converges uniformly
and almost surely to a non random profile depending only on the covariance and
the considered linear functional of the field. This is a significant
improvement of the weaker $L^2$-convergence in probability previously obtained
in the case of conditioning on a large quadratic functional.
