New determinations of eclipse timings were made of the eccentric eclipsing binaries DI Her, AS Cam, V380 Cyg, Y Cyg and V541 Cyg; these were determined from the APT photometry by Guinan and students James Marshall, Mark Devita and Jamison Maley. The apsidal motion rates have been measured for these stars by combining recently determined eclipse timings with those obtained earlier. Guinan is collaborating with Daniel Popper (UCLA) on determining the absolute properties of V380 Cyg, while the apsidal motion studies of most of the other stars is being done with Maloney (Villanova), Bradstreet (Eastern College), and Giménez (INTA, Spain). Recent apsidal motion determinations have been made by Guinan and students Marshall and Maley of DI Her and V541 Cyg. Interestingly, the recent determination of the rate of apsidal motion of deg/100yr for V541 Cyg is nearly 50 percent smaller than expected from general-relativistic and classical effects. This is noteworthy because in V541 Cyg the relativistic contribution to apsidal motion is nearly 5 times larger than the classical component, arising from the tidal and rotational distortions of the stars. This work is supported by NSF grant AST 93-15365.