Searches for a scalar top quark and a scalar bottom quark have been performed using a data sample of 438 pb−1 at centre-of-mass energies of √s = 192–209 GeV collected with the OPAL detector at LEP. No evidence for a signal was found. The 95% confidence level lower limit on the scalar top quark mass is 97.6 GeV if the mixing angle between the supersymmetric partners of the left- and right-handed states of the top quark is zero. When the scalar top quark decouples from the Z0 boson, the lower limit is 95.7 GeV. These limits were obtained assuming that the scalar top quark decays into a charm quark and the lightest neutralino, and that the mass difference between the scalar top quark and the lightest neutralino is larger than 10 GeV. The complementary decay mode of the scalar top quark decaying into a bottom quark, a charged lepton and a scalar neutrino has also been studied. The lower limit on the scalar top quark mass is 96.0 GeV for this decay mode, if the mass difference between the scalar top quark and the scalar neutrino is greater than 10 GeV and if the mixing angle of the scalar top quark is zero. From a search for the scalar bottom quark, a mass limit of 96.9 GeV was obtained if the mass difference between the scalar bottom quark and the lightest neutralino is larger than 10 GeV.