Phase transitions are ubiquitous in our three-dimensional world. By contrast most conventional transitions do not occur in infinite uniform two-dimensional systems because of the increased role of thermal fluctuations. Here we explore the dimensional crossover of Bose-Einstein condensation (BEC) for a weakly interacting atomic gas confined in a novel quasi-two-dimensional geometry, with a flat in-plane trap bottom. We detect the onset of an extended phase coherence, using velocity distribution measurements and matter-wave interferometry. We relate this coherence to the transverse condensation phenomenon, in which a significant fraction of atoms accumulate in the ground state of the motion perpendicular to the atom plane. We also investigate the dynamical aspects of the transition through the detection of topological defects that are nucleated in a quench cooling of the gas, and we compare our results to the predictions of the Kibble-Zurek theory for the conventional BEC second-order phase transition.