What he said plus to help hold the key with the Fiat symbol up and towards the button and angle the other end downward slightly and use the key to push in the button. I too looked in the US manual and somehow have overlooked it, but this is the procedure.Hold the dead key up against the start/stop button.
The process is described on page 13 of my Australian (Abarth) manual (my understanding is that the Fiat and/or U.S. manual page numbers differ).
Mind you, you ought never to get into such a situation because the car will issue a warning when it detects the key's battery is getting low (the key indicator light in the instrument cluster will flash for about 30 seconds after the engine is turned off).