While the city has won international praise for its efforts to stem the spread of the virus, its infections have been rising sharply in recent weeks, to 1,114 on Friday, and five people have died.
"We have decided that instead of tightening incrementally over the next few weeks, we should make a decisive move now, to pre-empt escalating infections," Lee said in a speech, and promised more support for households and businesses.
Food establishments, markets and supermarkets, clinics, hospitals, utilities, transport and key banking services will remain open.
The new measures will be in place from April 7 until May 4, while schools will move to full home-based learning from April 8. The measures could be extended beyond a month if the situation did not improve, authorities said.
The prime minister urged everyone to stay at home as much as possible and to avoid socialising with others beyond their own household.
"Go out only to do essential things," he said.
He said the country had enough food supplies to last through this period and beyond.
The moves will help reduce the risk of a big outbreak occurring, and it should also help to gradually bring numbers down, Lee said, which will then allow the authorities to relax some of the measures.
The Southeast Asian nation has adopted some social distancing measures to curb the spread of the virus, but had let schools, offices and restaurants remain open.
The city-state's finance minister last month unveiled more than $30 billion in new measures to help businesses and households.