If the clock for a college football game has one second left on it for the last play, does the whole play count?
Yes, in college and professional football, the clock does not stop a play. If the clock runs down to 0 while a play is going on, the quarter is over when the play is done.
In addition, if certain circumstances occur on the last play (such as specific penalties), the quarter will be extended by one more play.
From the NCAA Football 2013 and 2014 Rules and Interpretations, Rule 3 (Periods, Time Factors and Substitutions), Section 2, Article 1 (Length of Periods and Intermissions):
a. No period shall end until the ball is dead and the referee declares the period ended.
From the 2013 Official Playing Rules of the NFL, Rule 4, Section 8 (Extension of a Period or a Half):
Section 8 Extension of a Period or a Half
BALL IN PLAY
Article 1 If time expires at the end of any period while the ball is in play, the period continues until the down ends.