Magnetic resonance velocimetry (MRV) can measure all three components of a time varying velocity field but as the spatial resolution is increased the measurements become increasingly noisy. To acquire velocity images of acceptable signal-to-noise ratio, repeated scans are required, leading to long acquisition times. We present an algorithm that is capable of reconstructing magnetic resonance velocimetry signals from a single scan, by formulating a Bayesian inverse Navier-Stokes problem for the unknown velocity field. This allows us to estimate the uncertainties of the unknowns by approximating their posterior covariance with a quasi-Newton method. We first test the method for synthetic noisy images of 2D flows and observe that the method successfully reconstructs and segments the noisy synthetic images with a signal-to-noise ratio (SNR) of 3. Then an MRV experiment to acquire images of an axisymmetric flow for low ~6 and high (>30) SNRs. We show that the method is capable of reconstructing and segmenting the low SNR images, producing noiseless velocity fields and a smooth segmentation, with negligible errors compared with the high SNR images. This amounts to a reduction of the total scanning time by a factor of 27. At the same time, the method provides additional knowledge about the physics of the flow (e.g. pressure), and addresses the shortcomings of MRV (low spatial resolution and partial volume effects) that otherwise hinder the accurate estimation of wall shear stresses.