Batter hits fly ball down the third base line. The fielder, with both feet in fair territory, reaches into foul territory where the ball touches his glove. Is this a fair ball or a foul ball?

Answer: Foul ball.

Under Rule 2.00 of Official Baseball Rules, the definition of a foul ball is [emphasis added by me]:

A FOUL BALL is a batted ball that settles on foul territory between home and first base, or between home and third base, or that bounds past first or third base on or over foul territory, or that first falls on foul territory beyond first or third base, or that, while on or over foul territory,touches the person of an umpire or player, or any object foreign to the natural ground.
A foul fly shall be judged according to the relative position of the ball and the foul line, including the foul pole, and not as to whether the infielder is on foul or fair territory at the time he touches the ball.
Rule 2.00 (Foul Ball) Comment: A batted ball not touched by a fielder, which hits the pitcher’s rubber and rebounds into foul territory, between home and first, or between home and third base is a foul ball.