A search for pair-produced leptoquarks is performed using e+e- collision events collected by the OPAL detector at LEP at centre-of-mass energies between 189 and 209 GeV. The data sample corresponds to a total integrated luminosity of 596 pb^-1. The leptoquarks are assumed to be produced via couplings to the photon and the Z0. For a given search channel only leptoquark decays involving a single lepton generation are considered. No evidence for leptoquark pair production is observed. Lower limits on masses for scalar and vector leptoquarks are calculated. The results improve most of the LEP limits derived from previous searches for the pair production process by 10-25 GeV, depending on the leptoquark quantum numbers.