We study, in two-dimensional QCD and in the large-N c limit, the properties of the gauge invariant quark Green's function, defined with a path-ordered phase factor along a straight line. The analysis is done by means of an exact integrodifferential equation. The Green's function is found to be infrared finite, with singularities represented by an infinite number of threshold type branch points with a power −3/2, starting at positive mass squared values. Its expression is analytically determined.