A room temperature magnetoelectric multiferroic is of interest as, e.g., magnetoelectric random access memory. Bulk samples of the perovskite (1-x)BiTi(1-y)/2FeyMg(1-y)/2O3-xCaTiO3 (BTFM-CTO) are simultaneously ferroelectric, weakly ferromagnetic, and magnetoelectric at room temperature. In BTFM-CTO, the volatility of bismuth oxide, and the complex subsolidus reaction kinetics, cause the formation of a microscopic amount of ferrimagnetic spinel impurity, which complicates the quantitative characterization of their intrinsic magnetic and magnetoelectric properties. Here, a controlled synthesis route to single-phase bulk samples of BTFM-CTO is devised and their intrinsic properties are determined. For example, the composition x = 0.15, y = 0.75 shows a saturated magnetization of 0.0097 mu(B) per Fe, a linear magnetoelectric susceptibility of 0.19(1) ps m(-1), and a polarization of 66 mu C cm(-2) at room temperature. The onset of weak ferromagnetism and linear magnetoelectric coupling are shown to coincide with the onset of bulk long-range magnetic order by neutron diffraction. The synthesis strategy developed here will be invaluable as the phase diagram of BTFM-CTO is explored further, and as an example for the synthesis of other compositionally complex BiFeO3-related materials.