At the High Court in Glasgow, 48-year-old Paul McCallum, from Blantyre, Lanarkshire, was unanimously found guilty of sexually abusing the girl from the age of five.
McCallum also attempted to rape the girl.
McCallum was also found guilty of sexually abusing another girl on one occasion when she was aged between eight and nine.
The court heard that McCallum has a previous conviction from 1983 for under-age sex.
Judge Alastair Stewart QC remanded him in custody and said: "A custodial sentence in this case, I think, is inevitable."
McCallum will be sentenced next month.