Searches for a scalar top quark and a scalar bottom quark have been performed using a data sample of 182 pb−1 at a centre-of-mass energy of Full-size image (<1 K)189 GeV collected with the OPAL detector at LEP. No evidence for a signal was found. The 95% confidence level (C.L.) lower limit on the scalar top quark mass is 90.3 GeV if the mixing angle between the supersymmetric partners of the left- and right-handed states of the top quark is zero. In the worst case, when the scalar top quark decouples from the Z0 boson, the lower limit is 87.2 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. From a search for the scalar bottom quark, a mass limit of 88.6 GeV was obtained if the mass difference between the scalar bottom quark and the lightest neutralino is larger than 7 GeV. These limits significantly improve the previous OPAL limits.