Liverpool made several bids for the midfielder, but were unable to meet Villa's asking price for the England international.
Villa imposed a deadline for a deal to be agreed, which was not met, leading the club to announce Barry will not be leaving.
O'Neill is happy the situation has now reached a resolution and is looking forward to having Barry in his plans for the new season.
"Naturally we're all absolutely delighted that Gareth will be staying at Villa Park and relieved that there is resolution and closure to the transfer saga," O'Neill told the club's official website.
"At a very positive meeting on Monday it was agreed between Gareth, his agent, Mr Lerner and myself that a deadline should be communicated to Liverpool to finally resolve the situation.
"Liverpool were informed and that deadline came and passed.
"Now we can all look forward to the season ahead with relief, anticipation and ambition."