Republic Airways, dba Delta Connection, flight 5826 encountered unanticipated moderate turbulence while deviating around convective weather during climb-out from Jacksonville International Airport (JAX), Jacksonville, Florida en route to General Edward Lawrence Logan International Airport (BOS), Boston, Massachusetts. The turbulence occurred as the airplane passed through approximately 10,000 feet and resulted in a serious injury to a flight attendant (FA). Following the event, the flight diverted to Washington Dulles International Airport (IAD), Dulles, Virginia. Before the event flight, the flight crew had completed an inbound leg from BOS to JAX during which they experienced light turbulence during descent, between 6,000 ft and landing. Before departure, the captain conducted a standard crew briefing and advised the FAs that “light chop” was expected during climbout, with smoother conditions anticipated during cruise. The captain was the pilot flying (PF) for the return flight to Boston. During climbout, the captain observed cloud tops along the flight path at about 10,000 ft and believed the flight would skim the tops. However, the airplane passed through the cloud tops, and immediately encountered light chop, which intensified into moderate turbulence. The captain selected manual speed control and maintained 250 knots while climbing through the area. The first officer made the “10,000 ft” callout and turned off the sterile cockpit light (which informs the FAs that the flight is above 10,000 feet) prior to the onset of moderate turbulence. Shortly after the turbulence encounter, the “B” FA contacted the flight crew and reported that the “A” FA had fallen in the aft galley and appeared to be injured. The captain transferred airplane control and radio communications to the first officer so that she could handle the injury situation. The captain contacted dispatch and was subsequently connected to StatMD medical services, who advised they apply ice and administer acetaminophen from the Emergency Medical Kit. The flight crew and dispatch discussed if the flight should continue to BOS or divert to a closer airport. After determining that the “A” FA could not perform her duties due to her injuries, dispatch recommended the flight divert to IAD. After the injury, the “B” FA assumed the “A” FA’s duties in the forward cabin, and the injured FA remained seated in the aft jumpseat. Medical personnel met the airplane upon arrival at the gate and transported the injured FA to the hospital where she was diagnosed with a broken ankle.