When I connect the Arduino to my PC the Led "ON" lights for a second then turns OFF. I wonder if the Arduino has a fault?
Most likely a short between VCC (5 volt) and GROUND. When there is a short the fuse will kick in after a second, and disable the whole board, including the ON-led.
When you disconnect the board, the fuse will reset.
So either the board is faulty of the connection you made to it are.