We report on a novel contact scheme which for the first time makes possible magnetotransport measurements on individual quasi-1D quantum wires prepared by cleaved edge overgrowth. Contacting is achieved via two isolated 2D electron systems that are connected to one another by the quantum wire under study. Two-terminal magnetoresistance measurements along our modulation-doped quantum wires exhibit electron mobilities in excess of 2 × 105 cm2/Vs and show pronounced Shubnikov-de Haas oscillations. Clear evidence for confinement of the electrons in 2D is deduced from the retarded depopulation of Landau levels at low magnetic fields. The results are in close agreement with a calculation of the magnetic field dependence of the number of occupied subbands in a narrow channel.