In the giant cylindrical cells found in Characean algae, multitudes of the molecular motor myosin transport the cytoplasm along opposing spiralling bands covering the inside of the cell wall, generating a helical shear flow in the large central vacuole. It has been suggested that such flows enhance mixing within the vacuole (van de Meent, Tuval & Goldstein, Phys. Rev. Lett., vol. 101, 2008, paper no. 178102) and thereby play a role in regulating metabolism. For this to occur the membrane that encloses the vacuole, namely the tonoplast, must transmit efficiently the hydrodynamic shear generated in the cytoplasm. Existing measurements of streaming flows are of insufficient spatial resolution and extent to provide tests of fluid mechanical theories of such flows and information on the shear transmission. Here, using magnetic resonance velocimetry (MRV), we present the first measurements of cytoplasmic streaming velocities in single living cells. The spatial variation of the longitudinal velocity field in cross-sections of internodal cells of Chara corallina is obtained with spatial resolution of 16 μm and is shown to be in quantitative agreement with a recent theoretical analysis (Goldstein, Tuval & van de Meent, Proc. Natl. Acad. Sci. USA, vol. 105, 2008, p. 3663) of rotational cytoplasmic streaming driven by bidirectional helical forcing in the cytoplasm, with direct shear transmission by the tonoplast. These results highlight the open problem of understanding tonoplast motion induced by streaming. Moreover, this study suggests the suitability of MRV in the characterization of streaming flows in a variety of eukaryotic systems and for microfluidic phenomena in general.