We show that the one- and two-dimensional ideal Bose gases undergo a phase transition if the temperature is lowered at constant pressure. At the pressure-dependent transition temperature Tc (P) and in their thermodynamic limit the specific heat at constant pressure cp and the particle densityn diverge, the entropyS and specific heat at constant volumec v fall off sharply but continuously to zero, and the fraction of particles in the ground state N0/N jumps discontinuously from zero to one. This Bose-Einstein condensation provides a remarkable example of a transition which has most of the properties of a second-order phase transition, except that the order parameter is discontinuous. The nature of the condensed state is described in the large but finiteN regime, and the width of the transition region is estimated. The effects of interactions in real one- and two-dimensional Bose systems and recent experiments on submonolayer helium films are discussed briefly.