  The large diversity of exotic electronic phases displayed by two-dimensional
superconductors confronts physicists with new challenges. These include the
recently discovered quantum Griffith singularity in atomic Ga films,
topological phases in proximized topological insulators and unconventional
Ising pairing in transition metal dichalcogenide layers. In LaAlO3/SrTiO3
heterostructures, a gate tunable superconducting electron gas is confined in a
quantum well at the interface between two insulating oxides. Remarkably, the
gas coexists with both magnetism and strong Rashba spin-orbit coupling and is a
candidate system for the creation of Majorana fermions. However, both the
origin of superconductivity and the nature of the transition to the normal
state over the whole doping range remain elusive. Missing such crucial
information impedes harnessing this outstanding system for future
superconducting electronics and topological quantum computing. Here we show
that the superconducting phase diagram of LaAlO3/SrTiO3 is controlled by the
competition between electron pairing and phase coherence. Through resonant
microwave experiments, we measure the superfluid stiffness and infer the gap
energy as a function of carrier density. Whereas a good agreement with the
Bardeen-Cooper-Schrieffer (BCS) theory is observed at high carrier doping, we
find that the suppression of Tc at low doping is controlled by the loss of
macroscopic phase coherence instead of electron pairing as in standard BCS
theory. We find that only a very small fraction of the electrons condenses into
the superconducting state and propose that this corresponds to the weak filling
of a high-energy dxz/yz band, more apt to host superconductivity
