  Motivated by recent experiments on the organic superconductor
$\lambda-(BETS)_2FeCl_4$ we study the Jaccarino-Peter effect (JPE), the
earliest example of magnetic-field-induced superconductivity, from the point of
view of current-density-functional theory. It is found that both Meissner
(diamagnetic) and Pauli (paramagnetic) pair breaking are suppressed by an
exchange-correlation contribution to the vector potential, arising at the sites
of the magnetic ions. This explains a number of otherwise puzzling experimental
observations, and sheds new light on earlier theories of the JPE.
