The quantum effects of radiation pressure are expected to limit the sensitivity of second-generation gravitational-wave interferometers. Though ubiquitous, such effects are so weak that they have not been experimentally demonstrated yet. Using a high-finesse optical cavity and a classical intensity noise, we have demonstrated radiation-pressure induced correlations between two optical beams sent into the same moving mirror cavity. Our scheme can be used to retrieve weak correlations at the quantum level and has applications both in high-sensitivity measurements and in quantum optics.