We present a kinematic model of the neutral hydrogen in the spiral galaxy of the lens system PKS 1830-211, based on a Multi-Element Radio-Linked Interferometer Network (MERLIN) 1.4-GHz radio map and the integrated and redshifted 21-cm hydrogen absorption-line profile as measured with the Westerbork Synthesis Radio Telescope (WSRT). Degeneracies in the models do not allow a unique determination of the kinematic centre, and forthcoming deeper Hubble Space Telescope observations with the Advanced Camera for Surveys (ACS) are required to break this degeneracy. Even so, we measure the inclination of the hydrogen disc: i= 17 degrees-32 degrees, indicating a close to face-on spiral galaxy. The optical depth increases with radius over the extent of...