High-harmonic generation is a well-known method of producing coherent extreme-ultraviolet (EUV) light, with photon energies up to about 0.5 keV (refs 1, 2). This is achieved by focusing a femtosecond laser into a gas, and high harmonics of the fundamental laser frequency are radiated in the forward direction. However, although this process can generate high-energy photons, efficient high-harmonic generation has been demonstrated only for photon energies of the order 50-100 eV (ref. 5). Ionization of the gas prevents the laser and the EUV light from propagating at the same speed, which severely limits the conversion efficiency. Here we report a technique to overcome this problem, and demonstrate quasi-phase-matched frequency conversion of laser light into EUV. Using a modulated hollow-core waveguide to periodically vary the intensity of the laser light driving the conversion, we efficiently generate EUV light even in the presence of substantial ionization. The use of a modulated fibre shifts the energy spectrum of the high-harmonic light to significantly higher photon energies than would otherwise be possible. We expect that this technique could form the basis of coherent EUV sources for advanced lithography and high-resolution imaging applications. In future work, it might also be possible to generate isolated attosecond pulses.