Nineteen pillow basalts dredged within the rift valley of the Mid-Atlantic Ridge at36.8°N were studied by the Thellier stepwise heating method in order to determine the paleointensity of the geomagnetic field when they erupted on to the sea floor. Previously reported fission track ages are 2,000 to 6,000 years for the youngest rocks (mainly olivine basalts) and 10,000 to 100,000 years for the others (mainly plagioclase basalts and pyroxene basalts). All but three pillow basalts meet the conditions commonly considered as indicative of quite reliable paleointensity estimates: stability of the direction of NRM during its thermal demagnetization, constant ratio of NRM/TRM (natural remanent magnetization to thermoremanent magnetization) over 50% or more of the original NRM intensity (80 to 94% for 11 specimens), and reproducibility of low-temperature partial TRM (PTRM). However, strong field thermomagnetic measurements indicate that 11 of these 16 samples display a significant increase in Curie temperature (15 to 80°C) during the paleointensity experiments below 250°C, notwithstanding the linearity of the NRM-TRM plot in this temperature interval. This alteration, probably due to low-temperature oxidation of the specimens, seems typical of young pillow basalts and may result in paleointensity estimates which are too high. This result shows that excellence of remanence tests (NRM-TRM linearity and PTRM stability) does not ensure the absence of chemical changes during the Thellier experiments and therefore the validity of the paleointensity obtained. The assumption that reliable paleointensities are obtained when the Curie point increase is 10°C or less led to the selection of five of our specimens, all from the youngest group. Their mean paleointensity is 64.2±20.5 μT (standard deviation) and the corresponding virtual axial dipole moment (VADM) is 11.5±3.7× 10²² A m². Given the variations of the VADM of the Earth's magnetic field over the last 6,000 years, as established from archeomagnetic studies, our paleointensity results suggest that the latest eruptions on the inner floor of the Rift Valley at 36.8°N occurred 1,500±1,000 years ago.