  High-harmonic generation (HHG) is widely used for up-conversion of amplified
(near) infrared ultrafast laser pulses to short wavelengths. We demonstrate
that Ramsey-comb spectroscopy, based on two such pulses derived from a
frequency-comb laser, enables us to observe phase effects in this process with
a few mrad precision. As a result, we could perform the most accurate
spectroscopic measurement based on light from HHG, illustrated with a
determination of the $5p^6 \rightarrow 5p^5 8s~^2[3/2]_1$ transition at 110 nm
in $^{132}$Xe. We improve its relative accuracy $10^4$ times to a value of
$2.3\times10^{-10}$. This is 3.6 times better than shown before involving HHG,
and promising to enable $1S-2S$ spectroscopy of He$^+$ for fundamental tests.
