Lloyds said the offer would be available to institutional investors as well as intermediaries selling to ordinary retail shareholders.
The retail offering will contain an incentive allowing each retail investor to receive one free share for every 20 shares acquired, up to the value of £2,000, and held for a year after the float.
Lloyds' long-awaited announcement today comes after it was forced to dispose of more than 600 branches under EU rules on state aid - after it was itself rescued at the height of the financial crisis.
It spun the branches off into the revived TSB brand after a deal to sell them to the Co-op collapsed.
The group, which also includes Halifax and Bank of Scotland, is itself still 25% owned by the taxpayer.
Lloyds chief executive Antonio Horta-Osorio said: "The decision to proceed with an initial public offering of TSB is an important further step for the group as we act to meet our commitments to the European Commission.
"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."
TSB chief executive Paul Pester said: "Today is a significant milestone on our journey to create a major new competitive force in UK banking."
Mr Pester said he would expect 15%-20% of the shares on offer to go to retail investors. These would feature the "reward element" to persuade shareholders to hang on to their stock.
"It all depends on the level of demand from both the institutional investors and retail investors," he said.
Mr Pester warned that there would not be a "significant dividend in the early term", signalling no payout until the 2017 financial year.
But he said there was "strong appetite" from investors for the offer, 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.
The announcement comes at a time of uncertainty for stock market floats, with fashion retailer FatFace last week cancelling a share sale and over-50s focused holidays-to-insurance group Saga pricing its stock at the bottom of its expected range.
Mr Pester brushed off questions about how Lloyds would set the price in the TSB sale, saying: "That's a call for them."
The chief executive set out plans to grow the business which he described as a "high street bank not a Wall Street bank".
He said: "The TSB we see today is unlike any other UK retail bank: we have the mindset and growth potential of a challenger but with the scale and capabilities of an established player."
Mr Pester said the business was protected from mis-selling issues of the past such as the payment protection insurance (PPI) scandal.
Lloyds would be responsible for any claims up to the date of the share float and Mr Pester said TSB was determined to avoid the mistakes others had made, leaving a trail of past problems that continue to haunt them.
"We have no intention at all of creating the conduct tail that other banks have suffered from," he said.
TSB, which has 4.5 million customers, says that with 631 branches it is the seventh largest retail banking group in the UK. It plans to grow its balance sheet by 40%-50% over the next five years.
It is aiming to become a larger player in the current account market, growing from 4.2% to 6% over the next four to five years.
Mr Pester said it had 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.
Mr Pester played down the likelihood of imposing a loan-to-income cap on mortgages as recently imposed by parent group Lloyds, amid soaring house prices in London
He said last week's announcement, affecting those looking to borrow more than £500,000, was "very sensible" but that for TSB, less exposed to home lending in the capital, such a move would be considered "possibly, but not at this stage".