  We present a novel route to realizing topological superconductivity using
magnetic flux applied to a full superconducting shell surrounding a
semiconducting nanowire core. In the destructive Little-Parks regime, reentrant
regions of superconductivity are associated with integer number of phase
windings in the shell. Tunneling into the core reveals a hard induced gap near
zero applied flux, corresponding to zero phase winding, and a gapped region
with a discrete zero-energy state around one applied flux quantum, {\Phi}_0 =
h/2e, corresponding to 2{\pi} phase winding. Theoretical analysis indicates
that in the presence of radial spin-orbit coupling in the semiconductor, the
winding of the superconducting phase can induce a transition to a topological
phase supporting Majorana zero modes. Realistic modeling shows a topological
phase persisting over a wide range of parameters, and reproduces experimental
tunneling conductance data. Further measurements of Coulomb blockade peak
spacing around one flux quantum in full-shell nanowire islands shows
exponentially decreasing deviation from 1e periodicity with device length,
consistent with Majorana modes at the ends of the nanowire.
