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 the ground state of the system exhibits the complete Bose--Einstein condensation, and that the condensate is the unique minimizer of the 3D energy-critical nonlinear Schr\"odinger functional where the nonlinear term is coupled with the scattering energy of the interaction potential.