--- Writes through with SABMiller response, broker comment, share price ---
SABMiller said the US$104bn or £42.15 per share cash proposal from the producer of Stella Artois was inadequate.
The news follows SABMiller's rejection of two previous cash bids from ABI, the first at £38 per share and the second at £40 per share.
Shares in SABMiller rose 22.5p to 3644.5p. ABI's stock also gained 1.79%.
Broker Shore Capital reduced its recommendation on SABMiller to hold from buy at a target price of 3705p.
The broker's analyst Phil Carroll said: "We believe this represents a good deal for SABMiller shareholders and ultimately we expect a deal to be agreed, although the dance could continue for a while yet.
"Given the bounce in the share price today, we downgrade our buy recommendation to hold."
Earlier, ABI had said it was disappointed at SABMiller's rejection of its first two bids "without any meaningful engagement".
ABI said it believed its latest proposal should be highly attractive to SABMiller shareholders and provided an extremely compelling opportunity.
A merger would create a global beer company with revenue of US$64bn and underlying pre-tax earnings of US$24bn and a presence in virtually every major beer market.
ABI chief executive Carlos Brito said: "By bringing together our rich heritage, brands and people we would provide more opportunities for consumers to taste and enjoy the world's best beers.
"Put simply, we believe we can achieve more together than each of us could separately, bringing more beers to more people and enhancing value for all of our stakeholders."
On Tuesday, SABMiller said strong demand for brands such as Aguila Light and Poker in Colombia and Castle Lite in South Africa drove up beverage and lager volumes by 5%.
It brought forward its trading update for the six months to September 30 citing a need to ensure "timely release of trading information to the market", given news of the ABI interest.
Belgium's ABI - the world's biggest brewer - also makes Budweiser and Corona, while UK-based SABMiller - which holds second position - owns Peroni, Grolsch and Pilsner Urquell beers.
ABI employs about 1,000 people at its UK breweries in Magor, Samlesbury and Mortlake, as well as its company headquarters in Luton. It has at least 200 beer brands.
SABMiller has around 69,000 staff in more than 80 countries, from Australia, Zambia and Colombia to the Czech Republic, South Africa and the US, also producing more than 200 beers.
Formerly known as South African Breweries, SAB acquired US rival Miller Brewing Company in 2002, resulting in the renaming of the group as SABMiller.
A merged group would be worth about £177bn and would add AB InBev's dominance of Latin America to SABMiller's strong presence in Africa, as well as their breweries in Asia.