We study the phase separation phenomenon in the Ising model in
dimensions $d \geq 3$. To this end we work in a large box with plus boundary
conditions and we condition the system to have an excess amount of negative
spins so that the empirical magnetization is smaller than the spontaneous
magnetization $m^*$. We confirm the prediction of the phenomenological theory
by proving that with high probability a single droplet of the minus phase
emerges surrounded by the plus phase. Moreover, the rescaled droplet is
asymptotically close to a definite deterministic shape, the Wulff crystal,
which minimizes the surface free energy. In the course of the proof we
establish a surface order large deviation principle for the magnetization. Our
results are valid for temperatures $T$ below a limit of slab-thresholds
$\hat{T}_c$ conjectured to agree with the critical point $T_c$. Moreover, $T$
should be such that there exist only two extremal translation invariant Gibbs
states at that temperature, a property which can fail for at most countably
many values and which is conjectured to be true for every $T$. The proofs are
based on the Fortuin–Kasteleyn representation of the Ising model along
with coarse-graining techniques.To handle the emerging macroscopic objects we
employ tools from geometric measure theory which provide an adequate framework
for the large deviation analysis. Finally,we propose a heuristic picture that
for subcritical temperatures close enough to $T_c$, the dominant minus spin
cluster of the Wulff droplet permeates the entire box and has a strictly
positive local density everywhere.