  The predicted Lense-Thirring perihelion precession of Mercury induced by the
Sun's angular momentum through its general relativistic gravitomagnetic field
amounts to 2 milliarcseconds per century. It turned out to be compatible with
the latest experimental determinations of the supplementary perihelion
precession of Mercury with the INPOP15a ephemerides, whose accuracy level has
nowadays reached the magnitude of the predicted relativistic effect itself
thanks to the analysis of some years of tracking data of the MESSENGER
spacecraft, which orbited Mercury from 2011 to 2015. A dedicated analysis of
three years of MESSENGER data with the DE ephemerides allowed for a $25\%$
determination of the Sun's angular momentum by means of the Lense-Thirring
effect, which turned out to be highly correlated with the signature due to the
Solar quadrupole mass moment $J_2^{\odot}$.
