A B737 flight crew leaving their gate stopped abruptly to avoid a collision with another aircraft; which had been cleared to its gate and injured a Flight Attendant. The Controller misjudged the aircraft's taxi speed.

Narrative

After pushback from our gate; we started engines and were ready to taxi. ATC cleared us to taxi via Z; G; and Y. As we started to taxi; the First Officer and I cleared our respective sides and began to taxi. We were on Z and; as we started to approach G; we cleared left and right again. No other aircraft was seen and no information regarding any other traffic was given by ATC. We started our turn onto G to join Taxiway Y and; as we started to turn; the First Officer saw another aircraft approaching on Y with a high closure rate and he said; 'Stop. Stop.' I could not see what he was seeing as that area is blind from my seat. I quickly and firmly applied the brakes and brought the aircraft to a stop. At this time; the other aircraft called Ground and asked if they should give way to us and stopped as well. The Controller was confused and I have serious doubts that he even saw the conflict. At this time; I was contacted by the cabin and informed that a Flight Attendant had fallen in the cabin. We notified ATC that we were stopped and were not moving so we could check on the Crew in the cabin. The other aircraft continued to their gate. Our Flight crew informed me that the 'A' Flight Attendant had fallen; but was OK and the cabin was OK. We taxied out to 30 and I asked ATC what had happened with the taxi clearance. He admitted that we had not done anything wrong; the mistake was his; and that he had mistaken the distance between the aircraft. The Ground Controller had issued the other carrier a clearance to taxi to their gate after clearing the runway and also issued us a clearance to taxi for departure. He had misjudged the time it would take to get to the same space on the taxiway that he cleared us both to be on. The First Officer did a good job staying heads-up; spotting the conflict; and then made good; quick; and assertive communication.