That process should be sped up somewhat by the House bill if it passes. If the Senate can pass a standalone measure identical to the one coming out of the House, the bill can go straight to the president's desk for signature, thus making it more likely that repeal can get done in the final days of the lame duck session.
But the path to passage in the Senate is not assured. The legislative calendar in the chamber is jam-packed with legislation Democrats and the White House want to pass before the Republicans elected in November are sworn in next month. Senate leadership has said that repealing DADT is an important goal, but just one of many they have before the Senate adjourns.
In her tweet, Pelosi echoed the support she's given repeal throughout the entire Congressional session.
"Action on #DADT is long overdue," she wrote.