From a compound of a (as a negative particle) and phellos (in the sense of a stone as stubbing the foot); smoothness, i.e. (figuratively) simplicity -- singleness.
KJV: And they, continuing daily with one accord in the temple, and breaking bread from house to house, did eat their meat with gladness and singleness of heart,
NASB: Day by day continuing with one mind in the temple, and breaking bread from house to house, they were taking their meals together with gladness and sincerity of heart,