  Interfaces between complex oxides are emerging as one of the most interesting
playgrounds in condensed matter physics. In this special setting, in which
translational symmetry is artificially broken, a variety of novel electronic
phases can be promoted. Theoretical studies predict complex phase diagrams and
suggest the key role of the carrier density in determining the systems ground
states. A particularly fascinating system is the interface between the
insulators LaAlO$_{3}$ and SrTiO$_{3}$, which displays conductivity with high
mobility. Recently two possible ground states have been experimentally
identified: a magnetic state and a two dimensional (2D) superconducting
condensate. In this Letter we use the electric field effect to explore the
phase diagram of the system. The electrostatic tuning of the carrier density
allows an on/off switching of superconductivity and drives a quantum phase
transition (QPT) between a 2D superconducting state and an insulating state
(2D-QSI). Analyses of the magnetotransport properties in the insulating state
are consistent with weak localisation and do not provide evidence for
magnetism. The electric field control of superconductivity demonstrated here
opens the way to the development of novel mesoscopic superconducting circuits
