  We consider a trapped dilute gas of $N$ bosons in $\mathbb{R}^3$ interacting
via a three-body interaction potential of the form $N\, V(N^{1/2}(x-y,x-z))$.
In the limit $N\to \infty$, we prove that every approximate ground state of the
system is a convex superposition of minimizers of a 3D energy-critical
nonlinear Schr\"odinger functional where the nonlinear coupling constant is
proportional to the scattering energy of the interaction potential. In
particular, the $N$-body ground state exhibits complete Bose--Einstein
condensation if the nonlinear Schr\"odinger minimizer is unique up to a complex
phase.
