  We develop a complete theory of non-formal deformation quantization on the
cotangent bundle of a weakly exponential Lie group. An appropriate integral
formula for the star-product is introduced together with a suitable space of
functions on which the star-product is well defined. This space of functions
becomes a Frechet algebra as well as a pre-C*-algebra. Basic properties of the
star-product are proved and the extension of the star-product to a Hilbert
algebra and an algebra of distributions is given. A C*-algebra of observables
and a space of states are constructed. Moreover, an operator representation in
position space is presented. Finally, examples of weakly exponential Lie groups
are given.
