In this first of a series of papers describing polarimetric multifrequency Very Long Baseline Array (VLBA) monitoring of 3C 273 during a simultaneous campaign with the INTEGRAL gamma-ray satellite in 2003, we present 5 Stokes I images and source models at 7 mm. We show that a part of the inner jet (1-2 milliarcseconds from the core) is resolved in a direction transverse to the flow, and we analyse the kinematics of the jet within the first 10 mas. Based on the VLBA data and simultaneous single-dish flux density monitoring, we determine an accurate value for the Doppler factor of the parsec scale jet, and using this value with observed proper motions, we calculate the Lorentz factors and the viewing angles for the emission components in the jet. Our data indicates a significant velocity gradient across the jet with the components travelling near the southern edge being faster than the components with more northern path. We discuss our observations in the light of jet precession model and growing plasma instabilities.