Aims: This work presents a high--precision variability survey in the field of the old, super metal-rich open cluster NGC 6791. Methods: The data sample consists of more than 75,000 high-precision CCD time series measurements in the V band obtained mainly at the Canada-France-Hawaii Telescope, with additional data from S. Pedro Martir and Loiano observatories, over a time span of ten nights. The field covers an area of 42x28 arcmin^2. Results: We have discovered 260 new variables and re-determined periods and amplitudes of 70 known variable stars. By means of a photometric evaluation of the membership in NGC 6791, and a preliminary membership based on the proper motions, we give a full description of the variable content of the cluster and surrounding field in the range 16<V<23.5. Accurate periods can be given for the variables with P<4.0 d, while for ones with longer periods the limited time-baseline hampered precise determinations. We categorized the entire sample as follows: 6 pulsating, 3 irregular, 3 cataclysmic, 89 rotational variables and 61 eclipsing systems; moreover, we detected 168 candidate variables for which we cannot give a variability class since their periods are much longer than our time baseline. Conclusions: On the basis of photometric considerations, and of the positions of the stars with respect to the center of the cluster, we inferred that 11 new variable stars are likely members of the cluster, for 22 stars the membership is doubtful and 137 are likely non-members. We also detected an outburst of about 3 mag in the light curve of a very faint blue star belonging to the cluster and we suggest that this star could be a new U Gem (dwarf nova) cataclysmic variable. Comment: 24 pages, 19 Figures, A&A accepted