The hole-closing phenomenon is studied in a polymer Langmuir film with coexisting gaseous and liquid phases both as a test of hydrodynamic theories of a two-dimensional fluid embedded in a three-dimensional one and as a means to accurately determine line tension, an important parameter determining size, shape, and dynamics within these and other membrane model systems. The hole-closing curve consists of both it universal linear regime and a history-dependent nonlinear one. Improved experimental technique allows its to explore the origin of the nonlinear regime. The linear regime confirms previous theoretical work and yields a value lambda = (0.69 +/- 0.02) pN for the line tension of the boundary between the gaseous and liquid phases. The observed hole closing also demonstrates that the two-dimensional polymer gas must be taken its having, a small. probably negligible elasticity, so that line-tension measurements assuming that both phases are incompressible should be re-evaluated.