  We show -- both theoretically and experimentally -- that
Einstein-Podolsky-Rosen steering can be distilled. We present a distillation
protocol that outputs a perfectly correlated system -- the singlet assemblage
-- in the asymptotic infinite-copy limit, even for inputs that are arbitrarily
close to being unsteerable. As figures of merit for the protocol's performance,
we introduce the assemblage fidelity and the singlet-assemblage fraction. These
are potentially interesting quantities on their own beyond the current scope.
Remarkably, the protocol works well also in the non-asymptotic regime of few
copies, in the sense of increasing the singlet-assemblage fraction. We
demonstrate the efficacy of the protocol using a hyperentangled photon pair
encoding two copies of a two-qubit state. This represents to our knowledge the
first observation of deterministic steering concentration. Our findings are not
only fundamentally important but may also be useful for semi device-independent
protocols in noisy quantum networks.
