Laser-driven high-order harmonic generation 1,2 (HHG) provides tabletop sources of broadband extreme-ultraviolet (XUV) light with excellent spatial 3 and temporal 4 coherence. These sources are typically operated at low repetition rates, f rep 100 kHz, where phase-matched frequency conversion into the XUV is readily achieved 5,6 . However, there are many applications that demand the improved counting statistics or frequency-comb precision afforded by operation at high repetition rates, f rep > 10 MHz. Unfortunately, at such high f rep , phase matching is prevented by the accumulated steady-state plasma in the generation volume 7-11 , setting stringent limitations on the XUV average power. Here, we use gas mixtures at high temperatures as the generation medium to increase the translational velocity of the gas, thereby reducing the steady-state plasma in the laser focus. This allows phase-matched XUV emission inside a femtosecond enhancement cavity at a repetition rate of 77 MHz, enabling a record generated power of ∼2 mW in a single harmonic order. This power scaling opens up many demanding applications, including XUV frequency-comb spectroscopy 12,13 of few-electron atoms and ions for precision tests of fundamental physical laws and constants 14-20 .The highly-nonlinear HHG process requires peak laser intensities around 10 14 W/cm 2 , which necessitates large laser pulse energies 10 µJ, and short pulse durations 100 fs, as typically reached with low repetition rate, chirped-pulse amplified 21 laser systems. However, high repetition rates are desirable for applications such as photoelectron spectroscopy 22-24 and microscopy 25 as well as electron-ion coincidence spectroscopy 26,27 , which are limited by counting detection or space-charge effects to few XUV ionization events per shot. Most notably, precision frequency-comb spectroscopy 12,13 requires f rep 10 MHz in order to stabilize the comb. Recent efforts allowed HHG to be directly driven at f rep 1 MHz, using either the direct output of a high-power oscillator 22,28 or the coherent combination of several fibre amplifiers 29,30 . Achieving the necessary intensity for HHG with f rep 10 MHz requires lasers with average power in the kW range. Apart from one demonstration at 20 MHz, where the measured XUV power was extremely low 31 , higher repetition rates up to 250 MHz 32 have been facilitated only by using passive enhancement cavities, which store ∼10 kW of laser power, where a gas jet is introduced at an intracavity focus 7,10-12,33-35 .In a macroscopic extended medium, efficient HHG requires matching the phase velocities of the generating laser and the generated fields. This can be achieved by balancing neutral and plasma dispersion, the geometric phase shift due to focusing (the Gouy phase), and the HHG intrinsic dipole phase 5,36 . Achieving this balance becomes increasingly challenging as the repetition rate increases above ∼10 MHz. The reason for this difficulty is that the plasma generated by one pulse does not have time to clear the focal volume before t...