The party leader and his shadow housing minister Melanie Onn have made the pledge after it was revealed this week almost 600 homeless people died last year.
The official figures, published for the first time by the Office for National Statistics, showed a 24 per cent rise in deaths over five years.
It comes days after Gyula Remes, 43, died after collapsing outside the entrance to Parliament, the second rough sleeper living in the underpass by Westminster Tube station to die this year.
Labour want to remove the Vagrancy Act 1824 from the statute book, which makes it a crime to beg as well as sleep rough on Britain’s streets.
The party said it was used to bring a criminal charge nearly 3,000 times in 2016, with offences leading to fines of up to £1,000, and leaving those convicted with a two-year criminal record.
Ms Onn said: “It beggars belief that we still use Georgian-era laws to criminalise some of the most vulnerable in society. Treating rough sleepers as criminals does not solve the underlying causes of homelessness and makes it harder for them to access support to move away from the streets.
“Rather than criminalising rough sleepers Labour would support them, with 8,000 new homes available to those with a history of rough sleeping as part of a plan to eradicate rough sleeping within five years.”
And Mr Corbyn said: “It should shame us all that rough sleeping has doubled in the last eight years and nearly 600 people died while homeless last year. Homeless people need help, not punishment.”
It comes after Labour announced plans earlier this week for a £100 million cold weather fund to make emergency accommodation available for every rough sleeper during winter.
Crisis chief executive Jon Sparkes welcomed the proposal, saying: “It’s unfathomable that an act created nearly 200 years ago is still today being used to punish people who are sleeping rough – some of the most vulnerable members of our society – making an already devastating situation even more desperate.
“The Vagrancy Act is antiquated, outdated and cruel. Our research shows that criminalising rough sleepers does nothing to help resolve and tackle the causes of homelessness and is far more likely to prevent someone from accessing the vital services they need to move away from the streets.
“What we need is to treat people who are homeless with dignity and respect. We’re glad to see that politicians from across parties have acknowledged the need to scrap this law, including a commitment from the Labour Party, and efforts from Liberal Democrat MP Layla Moran to take a bill through parliament this year.
“Whilst alone it won’t end homelessness altogether, it is long overdue.”
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