Mass transport in fluidic channels under conditions of pressure-driven flow is controlled by a combination of convection and diffusion. For electrochemical measurements the height of a channel is typically of the same order of magnitude as the electrode dimensions, resulting in complex two- or three- dimensional concentration distributions. Electrochemical nanofluidic devices, however, can have such a low height-to-length ratio that they can effectively be considered as one-dimensional. This greatly simplifies the modeling and quantitative interpretation of analytical measurements. Here we study mass transport in nanochannels using electrodes in a generator-collector configuration. The flux of redox molecules is monitored amperometrically. We observe the transition from diffusion-dominated to convection-dominated transport by varying both the flow velocity and the distance between the electrodes. These results are described quantitatively by the one-dimensional Nernst-Planck equation for mass transport over the full range of experimentally accessible parameters.