This paper presents the optimal dimensional synthesis of a six-bar linkage with rotational constraints in which a point on the second dyad generates the desired path. The aim of the synthesis is to bring the generated path as close to the given path as possible, the given path being a combination of a rectilinear segment and a circular arc. A large population of mechanisms is formed using the method of controlled decrease of allowed deviations and the differential evolution algorithm - DE within the optimization procedure. Establishing the initial boundaries of allowed deviations "far" enough leads to significant accumulation of generated paths around the given one. The path is given by a large number of precision points. The mechanisms realizing satisfactory motion are obtained by the method of controlled decrease of allowed deviations with the application of specific adaptation of algorithm DE.