We present a new computational method for reconstructing a vector field from scattered, pulsed-wave ultrasound Doppler data. The main difficulty is that the Doppler measurements are incomplete, for they do only capture the velocity component along the beam direction. We thus propose to combine measurements from different beam directions. However, this is not yet sufficient to make the problem well posed because (i) the angle between the directions is typically small; and (ii) the data is noisy and nonuniformly sampled. We therefore introduce a vector-spline regularization that forces the solution to be well behaved while inducing some helpful coupling between the x and y components of the vector field. We discretize our vector-spline reconstruction problem in a uniform B-spline basis, obtaining a sparse system of equations that can be solved efficiently. We validate our method using real tissue Doppler data for which the ground truth is known. Finally, we present some patient results obtained with color Doppler imaging; in particular, we are able to observe blood-flow variations in the bifurcation of the carotid artery.