We present a three-dimensional simulation of the dynamic emergence of a twisted magnetic flux tube from the top layer of the solar convection zone into the solar atmosphere and corona. It is found that after a brief initial stage of flux emergence during which the two polarities of the bipolar region become separated and the tubes intersecting the photosphere become vertical, significant rotational motion sets in within each polarity. The rotational motions of the two polarities are found to twist up the inner field lines of the emerged fields such that they change their orientation into an inverse configuration (i.e., pointing from the negative polarity to the positive polarity over the neutral line). As a result, a flux rope with sigmoid-shaped, dipped core fields forms in the corona, and the center of the flux rope rises in the corona with increasing velocity as the twisting of the flux rope footpoints continues. The rotational motion in the two polarities is a result of propagation of nonlinear torsional Alfven waves along the flux tube, which transports significant twist from the tube's interior portion toward its expanded coronal portion. This is a basic process whereby twisted flux ropes are developed in the corona with increasing twist and magnetic energy, leading up to solar eruptions.