  We construct the first family of horizonless supergravity solutions that have
the same mass, charges and angular momenta as general supersymmetric rotating
D1-D5-P black holes in five dimensions. This family includes solutions with
arbitrarily small angular momenta, deep within the regime of quantum numbers
and couplings for which a large classical black hole exists. These geometries
are well-approximated by the black-hole solution, and in particular exhibit the
same near-horizon throat. Deep in this throat, the black-hole singularity is
resolved into a smooth cap. We also identify the holographically-dual states in
the N=(4,4) D1-D5 orbifold CFT. Our solutions are among the states counted by
the CFT elliptic genus, and provide examples of smooth microstate geometries
within the ensemble of supersymmetric black-hole microstates.
