The eclipsing white dwarf plus main-sequence binary NN Serpentis provides one of the most convincing cases for the existence of circumbinary planets around evolved binaries. The exquisite timing precision provided by the deep eclipse of the white dwarf has revealed complex variations in the eclipse arrival times over the last few decades. These variations have been interpreted as the influence of two planets in orbit around the binary. Recent studies have proved that such a system is dynamically stable over the current lifetime of the binary. However, the existence of such planets is by no means proven and several alternative mechanisms have been proposed that could drive similar variations. One of these is apsidal precession, which causes the eclipse times of eccentric binaries to vary sinusoidally on many year time-scales. In this Letter, we present timing data for the secondary eclipse of NN Ser and show that they follow the same trend seen in the primary eclipse times, ruling out apsidal precession as a possible cause for the variations. This result leaves no alternatives to the planetary interpretation for the observed period variations, although we still do not consider their existence as proven. Our data limit the eccentricity of NN Ser to e < 10−3. We also detect a 3.3 ± 1.0 s delay in the arrival times of the secondary eclipses relative to the best planetary model. This delay is consistent with the expected 2.84 ± 0.04 s Rømer delay of the binary, and is the first time this effect has been detected in a white dwarf plus M dwarf system.