Nanoscale mechanical oscillators(1) are used as ultrasensitive detectors of force(2), mass(3) and charge(4). Nanomechanical oscillators have also been coupled with optical and electronic resonators to explore the quantum properties of mechanical systems(5). Here, we report an optomechanical transducer in which a Si3N4 nanomechanical beam(6,7) is coupled to a disk-shaped optical resonator made of silica on a single chip. We demonstrate a force sensitivity of 74 aN Hz(-1/2) at room temperature with a readout stability better than 1% at the minute scale. Our system is particularly suited for the detection of very weak incoherent forces, which is difficult with existing approaches because the force resolution scales with the fourth root of the averaging time(8). By applying dissipative feedback(9) based on radiation pressure, we significantly relax this constraint and are able to detect an incoherent force with a force spectral density of just 15 aN Hz(-1/2) (which is 25 times less than the thermal noise) within 35 s of averaging time (which is 30 times less than the averaging time that would be needed in the absence of feedback). It is envisaged that our hybrid on-chip transducer could improve the performance of various forms of force microscopy(8,10-12).