Zhelev and Needham have recently created large, quasistable pores in artificial lipid bilayer vesicles. Initially created by electroporation, the pores remain open for up to several seconds before quickly snapping shut. This result is surprising, in light of the large line tension for holes in bilayer membranes and the rapid time scale for closure of large pores. We show how pores can be dynamically stabilized via a new feedback mechanism. We also explain quantitatively the observed sudden pore closure as a tangent bifurcation. Finally, we show how Zhelev and Needham's experiment can be used to measure accurately the pore line tension, an important material parameter. For their stearoyloleoylphosphatidylcholine/cholesterol mixture we obtain a line tension of 2.6 x 10(-6) dyn.