  We have experimentally investigated the formation of off-diagonal long-range
order in a gas of ultracold atoms. A magnetically trapped atomic cloud prepared
in a highly nonequilibrium state thermalizes and thereby crosses the
Bose-Einstein condensation phase transition. The evolution of phase coherence
between different regions of the sample is constantly monitored and information
on the spatial first-order correlation function is obtained. We observe the
growth of the spatial coherence and the formation of long-range order in real
time and compare it to the growth of the atomic density. Moreover, we study the
evolution of the momentum distribution during the nonequilibrium formation of
the condensate.
