Banco Sabadell and TSB have reached agreement on a £1.7 billion takeover by the Spanish bank.
The deal will see shareholders in the bank receive 340p per share.
Lloyds Banking Group has agreed to sell its 50% share.
TSB chief executive Paul Pester will continue in his current role.
TSB has received a takeover approach from Spain's Sabadell valuing the UK bank at £1.7 billion.
Lloyds Banking Group's sale of a 35% stake in TSB will net the taxpayer-backed bank £455m.
The Initial Public Offering (IPO) will see 30% of the stock allocated to around 60,000 ordinary retail ivestors as TSB returns to the market as an independent company for the first time since its 1995 merger with Lloyds.
TSB is currently the seventh biggest UK retail bank with 631 branches and the IPO valued its shares at 260p each, giving the company an overall value of £1.3bn.
Lloyds Bank's sale of 35% of TSB shares is an "important step" for the company, according to chief executive Antonio Horta-Osorio.
The sale of the shares follows a 2009 decision by the European Commission to force the sell-off of TSB in order to increase competition in the UK banking sector.
Lloyds will have to sell off the remainder of its shares in TSB by the end of next year as part of the ruling.
Mr Horta-Osorio welcomed the move, saying: "The successful initial public offering of TSB is an important further step for Lloyds Banking Group as we act to meet our commitments to the European Commission."
Lloyds Bank has increased the number of shares it is selling in TSB as a result of strong demand from investors.
The bank, a quarter of which is owned by the taxpayer, had initially decided to sell a 25% stake in TSB but has now upped the proportion to 35%.
The shares have been priced at 260p each, giving TSB a market value of £1.3bn.
TSB says it plans to grow its balance sheet by 40%-50% over the next five years.
The bank is aiming to become a larger player in the current account market, growing from 4.2% to 6% during that time.
Chief executive Paul Pester says it has already seen four to five as many people opening accounts every week since the TSB brand was re-launched last September than it had before.
Growth plans will also see TSB mortgages becoming available through brokers again from the start of next year.
TSB chief executive Paul Pester says there is "strong appetite" from investors for the flotation of TSB, with optimism in the UK and overseas over the strength of Britain's economic recovery.
Further tranches of TSB will be floated later, with Lloyds obliged to dispose of its remaining interest in the business by the end of 2015.
Details of the pricing of next month's offer have yet to be announced but reports put the book value of TSB at about £1.5 billion.