Any national postal strike would be the first since 1996
The Communications Workers Union (CWU) blamed the walkout on Royal Mail bosses not taking negotiations seriously, and said further strikes could follow.
Royal Mail chief executive Adam Crozier said the organisation was "very disappointed" and would do all it could to limit the impact of the walkout.
The dispute is over pay and the union's fears of potential job cuts.
The strike will be the first nationwide walkout at the Royal Mail since 1996.
Both sides have said they would welcome additional talks.
'Refusing to negotiate'
"We've tried our hardest to reach a negotiated settlement with the company, but the truth is again that Royal Mail are refusing to negotiate whatsoever on reaching a settlement before strike action takes place," said CWU deputy general secretary Dave Ward.
"We have also agreed further strike action within two weeks from 29 June that is designed to maximise the impact on Royal Mail but at minimum cost to the CWU members."
Mr Ward added that the strike would include Post Office workers as well as Royal Mail delivery staff.
In addition to turning down the Royal Mail's 2.5% pay offer, the CWU fears the Royal Mail's ongoing restructuring plans will cause up to 40,000 job cuts.
'Losing business'
The Royal Mail counters that it cannot afford to offer any more money and needs to make itself more efficient.
"We are losing business because we have failed to change and modernise - and as a result, our costs and therefore our prices are higher than those that rivals are charging in the intensely competitive business mail market, which makes up 90% of all postings," said Mr Crozier.
"That's the issue everyone in Royal Mail has to face and why we are so ready to continue talking with the union about the need to modernise."
CWU members at the Royal Mail voted for the strike action with a majority of 77% on a turnout of 60%.
Royal Mail postman delivering mail

The Communication Workers Union (CWU) has confirmed that Royal Mail workers are to hold a 24 hour strike on June 29, the first national post strike in the United Kingdom in 11 years.

In a statement, Dave Ward, CWU Deputy General Secretary, said that Royal Mail are "intent on cutting services, cutting jobs and cutting pay" and that "the Union remains committed to reaching an agreement" to resolve the issue. CWU members voted 77% in favour of taking strike action.

Royal Mail chief executive Adam Crozier said that the organisation was "very disappointed" to hear news of the announcement and explained that "we are losing business because we have failed to change and modernise."

Dave Ward of the CWU said that "We have tried to reach an agreement but Royal Mail are refusing to negotiate" and that he wanted to "put the record straight about the way that Royal Mail have been deliberately misleading the public on what this dispute is about. The Union has never asked for a 27% pay rise and we are not opposed in any shape or form to modernisation."