American Airlines flight 160 encountered turbulence during climb to cruise while enroute to Athens International Airport (ATH), Spata, Greece from O’Hare International Airport (ORD), Chicago, Illinois resulting in a flight attendant (FA) being seriously injured. The captain reported having completed a briefing with the cabin crew prior to the flight where he discussed turbulence precautions and pointed out areas specifically where they expected the flight would encounter turbulence. During climb to the cruise altitude of flight level (FL) 370 [37,000 feet], the flight crew reported that air traffic control (ATC) assigned an intermediate level off at FL 330. The captain and first officer stated that while at FL 330 they discussed the continual light chop that the flight was experiencing and decided to keep the fasten seat belt sign on. The flight was then cleared to climb to FL 350 and the crew stated that during the climb the ride was smooth. Soon after reaching FL 350, the flight crew reported answering a cabin call informing them that a service cart had injured a flight attendant (FA2). The flight crew then informed flight dispatch and diverted to Philadelphia International Airport (PHL), Philadelphia, Pennsylvania, where they made an overweight landing and were met by paramedics at the gate. The cabin crew reported that FA2 was standing in the back galley preparing the flight meals when the pilot announced on the public address (PA) system that they were going to experience some turbulence. When the turbulence was encountered, the flight had started to climb, and FA2 reported that in a matter of seconds she was pinned between the cart and the back galley wall. She added that the cart then began to fall, knocking her to the floor. She said the cart fell on her before she could get up and the back of her right ankle was impacted, causing a large tear in the skin and subsequent bleeding. An onboard passenger who was an acute care registered nurse practitioner assisted FA2. Upon landing, FA2 was removed from the aircraft on a stretcher and taken to a hospital where she was diagnosed with an Achilles tendon injury, as well as other injuries. The cabin crew stated that the cart’s brakes had been applied but thought it had malfunctioned causing the food cart to move and injure FA2. According to American Airlines, the vendor who maintains and repairs the carts was unable to identify the event cart nor locate its maintenance and repair history.