Chancellor Rishi Sunak stressed that the forecast - which assumes the lockdown continues for another two months - was "just one potential scenario", but said it was "important that we're honest" about the economic impact of the virus causing "hardship ahead".
The grim forecast intensified the debate about when the lockdown should be eased.
Dominic Raab, the acting prime minister, is expected to announce tonight NZT that it will carry on into next month, but many ministers are anxious to find ways of easing restrictions as soon as possible to ensure the long-term economic damage does not lead to more deaths than the virus itself.
The official UK death toll hit 12,107 yesterday as another 778 people died, with government sources stressing there could be no talk of an exit strategy "before we have reached the peak".
NHS England said the number of coronavirus hospital admissions was "stabilising", but deaths would take longer to fall significantly.
The economic hit from coronavirus has been far worse than ministers expected, costing billions more per week than predicted because more businesses have shut their doors and shed staff than was hoped.
The OBR, which is independent of the Government but works closely with the Treasury, said Gross Domestic Product could fall by more than a third in the second quarter of the year if the lockdown lasts for three months - into June - and was followed by another three months of partial lockdown.
It would be the deepest recession since records began, and would put the UK on course for the biggest annual decline in the economy since 1709.
Sunak said the effect of the virus was "unlike anything we have seen before" and that the OBR figures "suggest that the scale of what we are facing will have serious implications".
He added: "These are tough times and there will be more to come. People should know that there's hardship ahead. We won't be able to protect every job or every business."
While the OBR said the economy could quickly bounce back once restrictions were lifted, with output returning to normal by the end of the year, unemployment would take longer to recover.
It said the jobless total would rise by 2.1 million to 3.4 million by the end of June, going from the current rate of 3.9 per cent to 10 per cent, its highest level since 1993, and potentially higher than at any time since the 1980s.
The OBR said unemployment would not return below 4 per cent until 2023.