We define a new diffusive matrix model converging towards the β -Dyson Brownian motion for all β∈[0,2] that provides an explicit construction of β-ensembles of random matrices that is invariant under the orthogonal/unitary group. For small values of β, our process allows one to interpolate smoothly between the Gaussian distribution and the Wigner semi-circle. The interpolating limit distributions form a one parameter family that can be explicitly computed. This also allows us to compute the finite-size corrections to the semi-circle.