We present results on the static q¯q potential from high-statistics simulations on 164, 243×32, and 324 lattices, using the standard Wilson action at β=6.0,6.2, and 6.4 on the Connection Machine CM2. To decrease noise and increase accuracy, we applied a suitable local smoothing technique on the spatial parts of Wilson loop operators. As a result, we find a violation of asymptotic scaling of the string tension, as signaled by √σΛL=96.7(1.6)(2.6),86.4(1.0)(1.9),82.3(0.8)(1.7), for the three β values, with statistical and systematic errors. We observe a linear confining potential up to distances of 2 fm. A volume of (1.5fm)3 appears to be sufficient to avoid finite-size effects within our statistical accuracy (≅ 1%).