The recent theories of planetary formation lead to a gravitationally unstable structure of the proto-Earth in the accretion stage, which is composed of three layers: an innermost undifferentiated solid core, an intermediate metal-melt layer, and an outermost silicate-melt layer. Taking this configuration as an initial state, we investigate the Earth's core formation due to the Rayleigh-Taylor instability by using the quantitative results on the instability in a self-gravitating fluid sphere obtained from another paper (S. Ida, Y. Nakagawa, and K. Nakagawa, submitted). We find that the instability occurs through the translational mode on a time scale of about 10 hr if the thickness of the metal-melt layer ⪆1 km. This leads to the conclusion that the Earth's core began to form through the translation of the innermost undifferentiated solid core as soon as the outer layer was melted and differentiated in the late accretion stage. In addition, we examine the rotational effects of the instability; the translation occurs most often along the rotational axis. But this preference is weak, since the rotational energy is small compared to the gravitational one.