On the basis of first-principles Monte Carlo simulations we find that the screw dislocation along the hexagonal axis of an hcp 4He crystal features a superfluid (at T→0) core. This is the first example of a regular quasi-one-dimensional supersolid—the phase featuring both translational and superfluid orders, and one of the cleanest cases of a Luttinger-liquid system. In contrast, the same type of screw dislocation in solid H2 is insulating.