  We experimentally study the effect of disorder on trapped quasi
two-dimensional (2D) 87Rb clouds in the vicinity of the
Berezinskii-Kosterlitz-Thouless (BKT) phase transition. The disorder
correlation length is of the order of the Bose gas characteristic length scales
(thermal de Broglie wavelength, healing length) and disorder thus modifies the
physics at a microscopic level. We analyze the coherence properties of the
cloud through measurements of the momentum distributions, for two disorder
strengths, as a function of its degeneracy. For moderate disorder, the
emergence of coherence remains steep but is shifted to a lower entropy. In
contrast, for strong disorder, the growth of coherence is hindered. Our study
is an experimental realization of the dirty boson problem in a well controlled
atomic system suitable for quantitative analysis.
