A student pilot and an instructor lost control of the C177 when the brakes failed as they attempted to hold short of the runway. Examination determined the brake fluid level was inadequate to service both wheels.

Narrative

The Private Pilot Student had no problem landing at the airport and parking on the West ramp at the FBO. He even did a tight 360 degree turn to pull into a parking spot with no problem. Later; I taxied the plane (and the private pilot) to the East side to park at the airport restaurant. I asked the student if he had noticed the left brake pedal being a little softer than the right. He said 'no' he had not noticed that. Leaving the restaurant we were cleared to cross Runway XX from East to West at Taxiway G. The Tower asked us to expedite taxi and then amended clearance to hold short Runway XX at Gulf. The Private Pilot who was in the left seat pressed on the brakes and the aircraft veered 90 degrees to the right. It remained on the taxiway. The left brake had failed completely - the pedal now went all the way to the floor. We contacted the Tower to advise of our situation and to request assistance. The Tower asked if we had crossed the hold short line. We noted our wheels had not crossed. The Tower Controller stated that he thought the left wing tip had crossed the hold short line. He asked us to shutdown in our present position and await a tug. A tug pulled the aircraft to the FBO on the West ramp. Maintenance personnel serviced the aircraft and pumped quite a bit of brake fluid into the Left brake line. Then they pumped additional fluid into the right brake line. After servicing; the left brake worked fine.I first noticed a soft pedal while taxiing to the restaurant. If we followed up better we could have averted the incident on the taxiway. We could have had the Private Pilot pay special attention to any change in the condition of the brake since he had last felt it. I had flown earlier that day with another student; and neither student had reported a problem. So when I first noticed the left pedal being softer than the right I just assumed it had been that way all day long.