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 bank is to be valued at between £1.1 billion and £1.45 billion when Lloyds Banking Group sells a 25% stake in the bank towards the end of June.
ITV News' Business Editor Joel Hills tweeted:
Priced to go: Lloyds will sell shares in TSB at 220 pence to 290 pence - at mid-point, values bank at £1,275 million. Final price June 20th
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.
TSB is protected against compensation claims by its parent company, the bank's chief executive Paul Pester says.
TSB boss says it has "indemnity" against any claims for misconduct (PPI etc) until it lists. Lloyds would deal with compensation.
Misconduct claims for the missale of payment protection insurance (PPI) have cost the industry billions of pounds, according to estimates from the Financial Conduct Authority.
Lloyds chief executive Antonio Horta-Osario says TSB has "a strong balance sheet" and is well protected from the issues that caused the banking crisis.
TSB has a national network of branches, a strong balance sheet and significant economic protection against legacy issues.
It is already operating on the UK high street and is proving to be a strong and effective challenger, further enhancing competition in the UK banking sector.