Searches for scalar top and scalar bottom quarks have been performed at center-of-mass energies between 161 GeV and 183 GeV using the L3 detector at LEP. No signal is observed. Model-independent limits on production cross sections are determined for the two decay channels t ̃ 1 →c χ ̃ 1 0 and b ̃ 1 →b χ ̃ 1 0 . Within the framework of the Minimal Supersymmetric extension of the Standard Model mass limits are derived. For mass differences between t ̃ 1 and χ ̃ 1 0 greater than 10 GeV a 95% C.L. limit of 81.5 GeV is set on the mass of the Supersymmetric partner of the left-handed top. A supersymmetric partner of the left-handed bottom with a mass below 80 GeV is excluded at 95% C.L. if the mass difference between b ̃ 1 and χ ̃ 1 0 is greater than 20 GeV.