  In Part I (arXiv:1209.2045) we computed the Stokes data, though not the
"connection matrix", for the smooth solutions of the tt*-Toda equations whose
existence we established by p.d.e. methods. Here we give an alternative proof
of the existence of some of these solutions by solving a Riemann-Hilbert
problem. In the process, we compute the connection matrix for all smooth
solutions, thus completing the computation of the monodromy data. We also give
connection formulae relating the asymptotics at zero and infinity of all smooth
solutions, clarifying the region of validity of the formulae established
earlier by Tracy and Widom. Finally, for the tt*-Toda equations, we resolve
some conjectures of Cecotti and Vafa concerning the positivity of S+S^t (where
S is the Stokes matrix) and the unimodularity of the eigenvalues of the
monodromy matrix.
