The properties of the gauge invariant two-point quark Green's function are studied in the large-Nc limit of two-dimensional QCD. The analysis is done by means of an exact integrodifferential equation. The Green's function is found infrared finite, with singularities in the momentum squared variable represented by an infinite number of threshold type branch points with a power -3/2, starting at positive mass squared values, with cuts lying on the positive real axis. The expression of the Green's function is analytically determined.