We consider the system of $ N (\geq 2) $ hard balls with masses $ m_{1}, \ldots, m_{N} $ and radius r in the flat torus $ \mathbb{T}_{L}^{\nu} = \mathbb{R}^{\nu} / L \cdot \mathbb{Z}^{\nu} $ of size $ L, \nu \geq 3 $ . We prove the ergodicity (actually, the Bernoulli mixing property) of such systems for almost every selection $ (m_{1}, \ldots, m_{N}; L) $ of the outer geometric parameters. This theorem complements my earlier result that proved the same, almost sure ergodicity for the case $ \nu = 2 $ . The method of that proof was primarily dynamical-geometric, whereas the present approach is inherently algebraic.