Using MHD numerical experiments in three dimensions, we study the emergence of a bipolar magnetic region from the solar interior into a model corona containing a large-scale, horizontal magnetic field. An arch-shaped concentrated current sheet is formed at the interface between the rising magnetized plasma and the ambient coronal field. Three-dimensional reconnection takes place along the current sheet, so that the corona and the photosphere become magnetically connected, a process repeatedly observed in recent satellite missions. We show the structure and evolution of the current sheet and how it changes in time from a simple tangential discontinuity to a rotational discontinuity with no null surface. We find clear indications that individual reconnection events in this three-dimensional environment in the advanced stage are not one-off events, but instead take place in a continuous fashion, with each field line changing connectivity during a finite time interval. We also show that many individual field lines of the rising tube undergo multiple processes of reconnection at different points in the corona, thus creating photospheric pockets for the coronal field. We calculate global measures for the amount of subphotospheric flux that becomes linked to the corona during the experiment and find that most of the original subphotospheric flux becomes connected to coronal field lines. The ejection of plasma from the reconnection site gives rise to high-speed and high-temperature jets. The acceleration mechanism for those jets is akin to that found in previous two-dimensional models, but the geometry of the jets bears a clear three-dimensional imprint, having a curved-sheet appearance with a sharp interface to the overlying coronal magnetic field system. Temperatures and velocities of the jets in the simulations are commensurate with those measured in soft X-rays by the Yohkoh satellite.