Searches for scalar top and scalar bottom quarks, as well as for mass-degenerate scalar quarks of the first two families are performed at 189 GeV centre-of-mass energy with the L3 detector at LEP. No signals are observed. Model-independent limits on the scalar top production cross sections are determined for the decay modes and . For scalar quarks of the other flavours decays are considered. Within the framework of the Minimal Supersymmetric Standard Model mass limits are set at 95% C.L. for these particles. Indirect limits on the gluino mass are also derived.