The proposed deal with O2's Spanish owner Telefonica will give Three a market share of around 37.5% and two million users, compared with Vodafone's estimated 40%.
O2, which sponsors Dublin's main concert arena and the country's rugby team, has 28% of the Irish market, employs 850 staff and has almost 70 stores. Telefonica is selling the business to cut debt.
Three owner Hutchison Whampoa said it will now have the scale and strength to "compete even more aggressively", although consumers will be left with just three main mobile operators. The other player is Eircom.
Three said O2 will be rebranded over the next three years, but added it was too soon to say whether jobs will be cut or when the O2 name will disappear on key sponsorship deals.
Hutchison Whampoa, owned by Chinese billionaire Li Ka-shing, is paying 780 million euros (£665 million) up front, plus another 70 million euros (£60 million) depending on it hitting financial targets.
Robert Finnegan, chief executive of Three Ireland, said: "As an ambitious competitor in the market we have been looking at ways to increase customer numbers and grow market share, and acquisition was the natural next step.
"Our combined 37.5% market share gives us the scale and financial strength to compete even more aggressively in the market to the benefit of consumers. Our ability to invest coupled with the combined subscriber base will create new competitive dynamics in the Irish telecoms market."
Three has about 9% of the Irish market, 450 staff and more than 50 stores. The network will have combined revenues of 803 million euros (£685 million). Telefonica will use the proceeds to help meet its target of cutting debt to below 47 million euros (£40.1 billion) this year.