We have observed the two-proton radioactivity of the previously unknown (19)Mg ground state by tracking the decay products in-flight. For the first time, the trajectories of the 2p-decay products, (17)Ne+p+p, have been measured by using tracking microstrip detectors which allowed us to reconstruct the 2p-decay vertices and fragment correlations. The half-life of (19)Mg deduced from the measured vertex distribution is 4.0(15) ps in the system of (19)Mg. The Q value of the 2p decay of the (19)Mg ground state inferred from the measured p-p-(17)Ne correlations is 0.75(5) MeV.