  We demonstrate control of the electron number down to the last electron in
tunable few-electron quantum dots defined in catalytically grown InAs
nanowires. Using low temperature transport spectroscopy in the Coulomb blockade
regime we propose a simple method to directly determine the magnitude of the
spin-orbit interaction in a two-electron artificial atom with strong spin-orbit
coupling. Due to a large effective g-factor |g*|=8+/-1 the transition from
singlet S to triplet T+ groundstate with increasing magnetic field is dominated
by the Zeeman energy rather than by orbital effects. We find that the
spin-orbit coupling mixes the T+ and S states and thus induces an avoided
crossing with magnitude $\Delta_{SO}$=0.25+/-0.05 meV. This allows us to
calculate the spin-orbit length $\lambda_{SO}\approx$127 nm in such systems
using a simple model.
