  We study the interface of the Ising model in a box of side-length $n$ in
$\mathbb Z^3$ at low temperature $1/\beta$ under Dobrushin's boundary
conditions, conditioned to stay in a half-space above height $h$ (a hard
floor). Without this conditioning, Dobrushin showed in 1972 that typically most
of the interface is flat at height $0$. With the floor, for small $h$, the
model is expected to exhibit {\it entropic repulsion}, where the typical height
of the interface lifts off of $0$. Detailed understanding of the SOS model -- a
more tractable height function approximation of 3D Ising -- due to Caputo et
al., suggests that there is a single integer value $-h_n^* \sim -c\log n$ of
the floor height, delineating the transition between rigidity at height $0$ and
entropic repulsion.
  We identify an explicit $h_n^*=( c_\star+o(1))\log n$ such that, for the
typical Ising interface above a hard floor at $h$, all but an
$\epsilon(\beta)$-fraction of the sites are propelled to be above height $0$ if
$h < h_n^*-1$, whereas all but an $\epsilon(\beta)$-fraction of the sites
remain at height $0$ if $h\geq h_n^*$. Further, $c_\star$ is such that the
typical height of the unconditional maximum is $(2c_\star + o(1))\log n$; this
confirms scaling predictions from the SOS approximation.
