Steven McIvor, 32, was one of two knife-wielding thieves who robbed the Letterbox Bistro in Balerno Main Street.
They stole Â£2,000 and 5,300 euros on 26 February 2016.
McIvor, from Edinburgh, had denied taking part in the robbery during a trial at the High Court in the city but was found guilty.
His co-accused Eddie Moffat, 27, was acquitted of the robbery charge on a not proven verdict, and admonished for a dangerous driving offence he admitted after spending almost 11 months in prison on remand. He was banned from driving for three years.
The owner of the post office/bistro business Steven Carlyle, 55, said he had been talking to his son when he became aware of two men entering the premises.
He told the court: "Then the language got worse and they demanded money. They were demanding all the money from the till and the euros."
The thieves had scarves over their mouths and each was armed with a knife.
Mr Carlyle said he pressed an alarm that was supposedly inaudible but turned out to be audible. He said: "That's when they started getting very agitated."
One of the intruders became very aggressive and a knife was thrust through a gap in a Perspex screen.
Mr Carlyle said: "They were swearing all the time, telling us to 'hurry up, get the money, get the cash'."
The trial judge, Lord Kinclaven, deferred sentence on unemployed McIvor, who has previous convictions for theft, for the preparation of a background report. He was remanded in custody.
A an has been convicted of taking part in an armed raid on a post office on the outskirts of Edinburgh.