Barr, which dates back to 1875 and also makes Tizer and Rubicon, revealed it made a new proposal following formal notification earlier this week that its now lapsed £1.4 billion merger with Britvic had been given the all clear.
The firm, based in Cumbernauld, near Glasgow, said Britvic turned down the latest potential offer even though it was on "more favourable terms".
Ronnie Hanna, AG Barr chairman, said: "While we are disappointed that the opportunity to create significant value for both sets of shareholders has been rejected, the board of AG Barr has every reason to be confident of its position as a stand-alone company."
Hertfordshire-based Britvic had raised doubts over a deal after hinting on Tuesday that it was under no pressure to resurrect a merger, in spite of the long-awaited competition clearance being secured.
Chairman Gerald Corbett said a new chief executive in Simon Litherland and the prospect of £30 million of cost savings over the next three years meant the merger benefits are "materially less than they were''.
Mr Corbett has been fiercely critical of the OFT's decision to refer the merger, which would have created one of the leading soft drinks companies in Europe.
In its final report, the commission said the brands owned by Britvic and AG Barr were not close competitors and consumers would not lose out.
Britvic claimed the new proposal from Barr, which would have seen the enlarged group made up of 65% Britvic and 35% Barr, was rejected as it represented only a "small improvement" on the previous offer. Mr Corbett added: "We wish Barr and its management team well. They are good people with a fine business."