Precision spectroscopy of the 2S − 4P1/2 and 2S − 4P3/2 transitions in atomic hydrogen is performed with a reproducibility of a few parts in 1012. A cryogenic beam of metastable 2S atoms is obtained by optical excitation, avoiding excessive heating of electron impact excitation used in all previous experiments of this kind. Despite the low temperature of 5.8 K, the first-order Doppler effect is the dominating systematic shift, which is suppressed to a very high degree. The effectiveness of this suppression is verified by employing a time-resolved detection scheme. This experiment should contribute to an improved determination of the Rydberg constant and the proton r.m.s. charge radius.