Context. CoRoT, the pioneer space-based transit search, steadily provides thousands of high-precision light curves with continuous time sampling over periods of up to 5 months. The transits of a planet perturbed by an additional object are not strictly periodic. By studying the transit timing variations (TTVs), additional objects can be detected in the system. Aims: A transit timing analysis of CoRoT-1b is carried out to constrain the existence of additional planets in the system. Methods: We used data obtained by an improved version of the CoRoT data pipeline (version 2.0). Individual transits were fitted to determine the mid-transit times, and we analyzed the derived O-C diagram. N-body integrations were used to place limits on secondary planets. Results: No periodic timing variations with a period shorter than the observational window (55 days) are found. The presence of an Earth-mass Trojan is not likely. A planet of mass greater than ~1 Earth mass can be ruled out by the present data if the object is in a 2:1 (exterior) mean motion resonance with CoRoT-1b. Considering initially circular orbits: (i) super-Earths (less than 10 Earth-masses) are excluded for periods less than about 3.5 days; (ii) Saturn-like planets can be ruled out for periods less than about 5 days; (iii) Jupiter-like planets should have a minimum orbital period of about 6.5 days. Based on observations obtained with CoRoT, a space project operated by the French Space Agency, CNES, with participation of the Science Programs of ESA, ESTEC/RSSD, Austria, Belgium, Brazil, Germany, and Spain.