We report the results of a search for D0-D0 mixing in D0 --> K+ pi- decays based on 400 fb(-1) of data accumulated by the Belle detector at KEKB. Both assuming CP conservation and allowing for CP violation, we fit the decay-time distribution for the mixing parameters x and y, as well as for the parameter R(D), the ratio of doubly Cabibbo-suppressed decays to Cabibbo-favored decays. The 95% confidence level region in the (x'2,y') plane is obtained using a frequentist method. Assuming CP conservation, we find x'2 < 0.72 x 10(-3) and -9.9 x 10(-3) < y' < 6.8 x 10(-3) at the 95% confidence level; these are the most stringent constraints on the mixing parameters to date. The no-mixing point (0,0) has a confidence level of 3.9%. Assuming no mixing, we measure R(D) = (0.377 +/- 0.008 +/- 0.005)%.