There have been previous hints that the transiting planet WASP-3b is accompanied by a second planet in a nearby orbit, based on small deviations from strict periodicity of the observed transits. Here we present 17 precise radial velocity (RV) measurements and 32 transit light curves that were acquired between 2009 and 2011. These data were used to refine the parameters of the host star and transiting planet. This has resulted in reduced uncertainties for the radii and masses of the star and planet. The RV data and the transit times show no evidence for an additional planet in the system. Therefore, we have determined the upper limit on the mass of any hypothetical second planet, as a function of its orbital period. Partly based on (1) observations made at the Centro Astronómico Hispano Alemán (CAHA), operated jointly by the Max-Planck Institut für Astronomie and the Instituto de Astrofísica de Andalucía (CSIC), (2) data collected with telescopes at the Rozhen National Astronomical Observatory, and (3) observations obtained with telescopes of the University Observatory Jena, which is operated by the Astrophysical Institute of the Friedrich-Schiller-University.