  Anionic DMPA monolayers spread on LaCl$_3$ solutions reveal strong cation
adsorption and a sharp transition to surface overcharging at unexpectedly low
bulk salt concentrations. We determine the surface accumulation of La$^{3+}$
with anomalous x-ray reflectivity and find that La$^{3+}$ compensates the lipid
surface charge by forming a Stern layer with $\approx 1$ La$^{3+}$ ion per 3
lipids below a critical bulk concentration, $c_t \approx 500 \mathrm{nM}$.
Above $c_t$, the surface concentration of La$^{3+}$ increases to a saturation
level with $\approx 1$ La$^{3+}$ per lipid, thus implying that the total
electric charge of the La$^{3+}$ exceeds the surface charge. This overcharge is
observed at $\approx$ 4 orders of magnitude lower concentration than predicted
in ion-ion correlation theories. We suggest that transverse electrostatic
correlations between mobile ions and surface charges (interfacial Bjerrum
pairing) may account for the charge inversion observed in this dilute regime.
