We construct a random matrix model for the bijection Ψ between clas- sical and free infinitely divisible distributions: for every d≥1, we associate in a quite natural way to each *-infinitely divisible distribution μ a distribution ℙdμ on the space of d×d Hermitian matrices such that ℙdμ*ℙdν=ℙdμ*ν. The spectral distribution of a random matrix with distribution ℙdμ converges in probability to Ψ(μ) when d tends to +∞. It gives, among other things, a new proof of the almost sure convergence of the spectral distribution of a matrix of the GUE and a projection model for the Marchenko–Pastur distribution. In an analogous way, for every d≥1, we associate to each *-infinitely divisible distribution μ, a distribution $\mathbb{L}_{d}^{\mu}$ on the space of complex (non-Hermitian) d×d random matrices. If μ is symmetric, the symmetrization of the spectral distribution of |Md|, when Md is $\mathbb{L}_{d}^{\mu}$ -distributed, converges in probability to Ψ(μ).