In stars with temperatures above 20*10^6 K, hydrogen burning is dominated by the CNO cycle. Its rate is determined by the slowest process, the 14N(p,gamma)15O reaction. Deep underground in Italy's Gran Sasso laboratory, at the LUNA 400 kV accelerator, the cross section of this reaction has been measured at energies much lower than ever achieved before. Using a windowless gas target and a 4pi BGO summing detector, direct cross section data has been obtained down to 70 keV, reaching a value of 0.24 picobarn. The Gamow peak has been covered by experimental data for several scenarios of stable and explosive hydrogen burning. In addition, the strength of the 259 keV resonance has been remeasured. The thermonuclear reaction rate has been calculated for temperatures 90 - 300 *10^6 K, for the first time with negligible impact from extrapolations.