  Interstitials and vacancies in the Abrikosov phase of clean Type II
superconductors are line imperfections, which cannot extend across macroscopic
equilibrated samples at low temperatures. We argue that the entropy associated
with line wandering nevertheless can cause these defects to proliferate at a
sharp transition which will exist if this occurs below the temperature at which
the crystal actually melts. Vortices are both entangled and crystalline in the
resulting ``supersolid'' phase, which in a dual ``boson'' analog system is
closely related to a two-dimensional quantum crystal of He$^4$ with
interstitials or vacancies in its ground state. The supersolid {\it must} occur
for $B\gg B_\times$, where $B_\times$ is the decoupling field above which
vortices begin to behave two-dimensionally. Numerical calculations show that
interstitials, rather than vacancies, are the preferred defect for $B\gg
\phi_0/\lambda_\perp^2$, and allow us to estimate whether proliferation also
occurs for $B\,\lot\,B_\times$.The implications of the supersolid phase for
transport measurements, dislocation configurations and neutron diffraction are
discussed.
