The ability to amplify optical signals is of pivotal importance across science and technology typically using rare-earth-doped fibres or gain media based on III-V semiconductors. A different physical process to amplify optical signals is to use the Kerr nonlinearity of optical fibres through parametric interactions(1,2). Pioneering work demonstrated continuous-wave net-gain travelling-wave parametric amplification in fibres(3), enabling, for example, phase-sensitive (that is, noiseless) amplification(4), link span increase(5), signal regeneration and nonlinear phase noise mitigation(6). Despite great progress(7-15), all photonic integrated circuit-based demonstrations of net parametric gain have necessitated pulsed lasers, limiting their practical use. Until now, only bulk micromachined periodically poled lithium niobate (PPLN) waveguide chips have achieved continuous-wave gain(16,17), yet their integration with silicon-wafer-based photonic circuits has not been shown. Here we demonstrate a photonic-integrated-circuit-based travelling-wave optical parametric amplifier with net signal gain in the continuous-wave regime. Using ultralow-loss, dispersion-engineered, metre-long, Si3N4 photonic integrated circuits(18) on a silicon chip of dimensions 5 x 5 mm(2), we achieve a continuous parametric gain of 12 dB that exceeds both the on-chip optical propagation loss and fibre-chip-fibre coupling losses in the telecommunication C band. Our work demonstrates the potential of photonic-integrated-circuit-based parametric amplifiers that have lithographically controlled gain spectrum, compact footprint, resilience to optical feedback and quantum-limited performance, and can operate in the wavelength ranges from visible to mid-infrared and outside conventional rare-earth amplification bands.