We study the Heisenberg antiferromagnet on a triangular lattice with √3 × √3-type distortion by numerical-diagonalization method. We obtain the zero-temperature magnetization process and the temperature dependence of the magnetic susceptibility. Our results suggest that, between the nondistorted triangular-lattice and dice-lattice systems, a nontrivial intermediate phase appears, where a spontaneous magnetization occurs but its magnitude is smaller than one-third of the saturated magnetization. In the small-distortion region toward the spontaneously magnetized phase, the magnetic susceptibility shows a two-step behavior of its peak structure.