She said she still hoped that MPs would approve her exit plan but admitted the process needed to implement this "will clearly not be completed before 29 March 2019" - the date Britain was due to leave the EU.
It comes after MPs rejected her deal for a second time last week and also voted to take a no-deal outcome off the table.
Ms May's decision to seek only a three-month extension to the Article 50 process was condemned by pro-EU MPs, who had demanded a longer delay to ensure the UK does not leave the bloc without an agreement.
The prime minister had previously said a three-month delay would be sufficient if a deal was passed by MPs, but that a longer extension would be needed if the Commons could not agree an exit plan.
But despite no withdrawal agreement having been approved, she decided to only seek a short delay after a backlash from Eurosceptic Tory MPs and cabinet ministers.
Her request will be discussed by EU leaders at a European Council summit in Brussels on Thursday. Ms May is expected to be asked to explain how she would use the extension to ensure parliament approves an exit deal.
Writing to Mr Tusk, the prime minister said the government's policy "remains to leave the European Union in an orderly manner on the basis of the withdrawal agreement and political declaration agreed in November".
Referring to the need for the UK to elect MEPs in elections this summer if Brexit is delayed beyond June, she said: "I do not believe that it would be in either of our interests for the UK to hold European Parliament elections."
The prime minister said she had planned to hold a third "meaningful vote" on her deal this week but was blocked by John Bercow, the Commons speaker, ruling that she would not be allowed to do so unless the agreement had changed "substantially".
Announcing her decision in the Commons at the start of Prime Minister's Questions, Ms May told MPs: "The government intends to bring forward proposals for a third meaningful vote.
"If that vote is passed, the extension will give the House time to consider the Withdrawal Agreement Bill. If not, the House will have to decide how to proceed.
"But as prime minister, I am not prepared to delay Brexit any further than June 30."
Her decision was criticised by pro-EU MPs.
Labour's Yvette Cooper, chair of the Commons home affairs committee, wrote on Twitter: "Truly shocking. This is a prime minister in the worst state of denial, refusing to listen to anyone, just still doing the same thing again and again, no plan B, heading stubbornly towards the cliff edge."
Some Tory Brexiteers, meanwhile, said Ms May was wrong to seek any extension at all.
Wellingborough MP Peter Bone told her: "If you continue to apply for an extension to Article 50 you will be betraying the British people. If you don't, you will be honouring their instruction.
"Prime minister, it is entirely down to you. History will judge you at this moment. Prime minister, which is it to be?"