The radial growth of a cholesteric liquid crystal phase was studied experimentally in a thick Hele-Shaw cell. In the experiments, nucleation and growth of the cholesteric phase occurs after a temperature quench from the isotropic liquid. The growth process changes from three- to two-dimensional when the growing phase touches the substrates of the Hele-Shaw cell. It was found that the growth rate generally increases at the dimensional crossover. Also, this increase in growth rate is approximately constant, independent of the depth of the temperature quench. Possible causes for the observed experimental behavior are discussed.