A pilot experiment suitable to test Newton's law of gravity down to the regime of acceleration typical of galaxies has been carried out in Omega Centauri. Stars in the extreme periphery of this globular cluster are used as test particles immersed in such weak gravitational field. The stellar velocity dispersion is found to remain constant at large radii, rather than decrease monotonically, starting at acceleration a=10e-7 cm/s2. This is comparable to the acceleration at which the effect of dark matter becomes relevant in galaxies. Explanations for this result within Newtonian dynamics exist (e.g. cluster evaporation, tidal effects, presence of dark matter) but require fine tuning of the relevant parameters in order to make the dispersion profile flat. An interesting alternative is that this result, together with a similar one for Palomar 13 and the anomalous behavior of spacecrafts outside the solar system, suggests a breakdown of Newton's law in the weak acceleration regime.