  In this second part, we study the Diophantine properties of values of
arithmetic Gevrey series of non-zero order at algebraic points. We rely on the
fact, proved in the first part, that the minimal differential operator (with
polynomial coefficients) which annihilates such a series has no non-trivial
singularity outside the origin and infinity.
  We show how to draw from this fact some transcendence properties, and recover
in particular the fundamental theorem of the Siegel-Shidlovsky theory on
algebraic independence of values of E-functions. The paradox of the title
points out the contrast between the qualitative aspect of this new argument and
the essentially quantitative aspect of the traditional approach.
  At last, we discuss q-analogues of the theory (theta-functions,
q-exponential,...).
