We present an experimental and modeling study of the effects of the 29 March 2006 solar eclipse in the topside ionosphere. Measurements of the densities and temperatures of the thermal electrons and ions were provided by instruments aboard the Centre National d'Etudes Spatiales microsatellite DEMETER, which flew over Europe and Africa near the time of maximum solar obscuration. Data from several orbits, either on the same day or on days encompassing the eclipse day, were available to determine a reference state of the ionosphere along the orbit in absence of eclipse. The comparison between this latter and the actual observations along the eclipse orbit reveal a clear thermal effect with a fast drop of about 200 K of the electron and ion temperatures that follows the variations of the solar UV flux in the F region of the ionosphere conjugate to the satellite position. The plasma density decreases by about 30% but with a significant delay and is better correlated with the solar UV flux averaged over the previous 1 to 2 h in the conjugate F region. This delayed and prolonged decrease of density induces an increase of the electron temperature to be higher than the reference ionosphere. We have also performed a modeling of the ionosphere using the SAMI2 code, after having introduced adequate modifications to reproduce fairly realistic eclipse conditions. Applied to the DEMETER conditions of observation, the model reproduces the observations very well. This work shows that the plasma temperature responds very quickly along the magnetic field lines to the variations of the energy available from the photoelectrons while the plasma density variations are controlled by more complex and slower transport processes.