  We apply a new and mathematically rigorous method for the quantization of
constrained systems to two-dimensional gauge theories. In this method, which
quantizes Marsden-Weinstein symplectic reduction, the inner product on the
physical state space is expressed through a certain integral over the gauge
group. The present paper, the first of a series, specializes to the Minkowski
theory defined on a cylinder. The integral in question is then constructed in
terms of the Wiener measure on a loop group. It is shown how $\th$-angles
emerge in the new method, and the abstract theory is illustrated in detail in
an example.
