But they said it was only a first step toward healing wounds left by the tests.
Helene Luc, a former senator who fought for official compensation, said: "The fact of having a draft law is the first victory.''
Michel Leger, president of the Association of Veterans of Nuclear Tests, said: "This is a step forward that we are greeting with satisfaction."
France carried out 210 tests in the Sahara Desert and the South Pacific, including Mururoa and Fangataufa, French Polynesian islands, from 1960 to 1996.
Pierre Leroy, a French army veteran, said he was present when a nuclear test misfired in the Sahara in 1962.
"We were 19, 20 years old. They told us, 'There are no risks, it's not dangerous'. There were no precautions," he said.
Staff who took part in the French tests, as well as residents of areas close to the testing zones, have long complained of health consequences including leukemia and other forms of cancer.
Morin defended the need for the tests at the time when France was building up its nuclear arsenal during the Cold War.
He said the explosions "allowed us to obtain an independent force of dissuasion, guaranteeing the protection of our vital interests and allowing us to be a power respected in the world alongside the other permanent members of the UN Security Council".
The draft law will be presented to parliament next month, with victims' groups pushing to add amendments to broaden the number of people who will be eligible for payouts.