We present a novel method for determining the total matter surface density of the Galactic disc by analysing the kinematics of a dynamically cold stellar stream that passes through or close to the Galactic plane. The method relies on the fact that the vertical component of energy for such stream stars is approximately constant, such that their vertical positions and vertical velocities are interrelated via the matter density of the Galactic disc. By testing our method on mock data stellar streams, with realistic phase-space dispersions and Gaia uncertainties, we demonstrate that it is applicable to small streams out to a distance of a few kilo-parsec, and that the surface density of the disc can be determined to a precision of 6 per cent. This method is complementary to other mass measurements. In particular, it does not rely on any equilibrium assumption for stars in the Galactic disc, and also makes it possible to measure the surface density to good precision at large distances from the Sun. Such measurements would inform us of the matter composition of the Galactic disc and its spatial variation, place stronger constraints on dark disc substructure, and even diagnose possible non-equilibrium effects that bias other types of dynamical mass measurements.