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Ramsey, the former Arsenal midfielder who joined Rangers on loan from Juventus in January, was brought on in extra time to take a penalty and was consoled by his teammates after his decisive miss.
Joe Aribo broke the deadlock in the 57th minute to put Rangers in front with a cool finish past Kevin Trapp from close range.
But Frankfurt pulled level just 12 minutes later as Rafael Borre moved swiftly to get on the end of Filip Kostic’s cross to steer the ball past Allan McGregor.
Frankfurt grew in confidence after their equaliser but Rangers held firm to force extra time.
Rangers had the best chance to win it in extra time as Kemar Roofe squared a ball across Frankfurt’s box which was met by Ryan Kent, however, a brilliant save by Trapp kept the score at 1-1 before the penalty shootout.
James Tavernier, Steven Davis, Scott Arfield all scored their penalties before Ramsey’s effort was saved by Trapp.
Roofe scored Rangers’ fifth effort but Borre’s brilliant penalty sealed the win for Frankfurt.
Giovanni van Bronckhorst’s side still have a chance to win silverware this season as they play Hearts in the Scottish Cup final at Hampden Park on Saturday.
Frankfurt, who finished 11th in the Bundesliga this season, have now won their first major European trophy since their success in the UEFA Cup in 1980.
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