We report on the laser-driven generation of purely neutral, relativistic electron-positron pair plasmas. The overall charge neutrality, high average Lorentz factor ($\gamma_{e/p} \approx 15$), small divergence ($\theta_{e/p} \approx 10 - 20$ mrad), and high density ($n_{e/p}\simeq 10^{15}$cm$^{-3}$) of these plasmas open the pathway for the experimental study of the dynamics of this exotic state of matter, in regimes that are of relevance to electron-positron astrophysical plasmas.