Boeing achieved a major milestone on Tuesday as the Federal Aviation Administration approved its plan to test fixes for the battery problems that have grounded its 787 jets since mid-January.

Boeing’s new battery design includes better protection in case a battery overheats. The F.A.A. could still demand changes if problems develop in the laboratory and flight tests. While Boeing hopes to begin fitting its redesigned batteries in the grounded 787 fleet by mid- to late April and resume commercial flights quickly after that, government officials are not sure the process will move that fast.

Still, the decision to start the tests is a big step in Boeing’s efforts to put the innovative jets back in the air. The 50 787s delivered to airlines have been grounded since mid-January after two planes developed battery problems — one battery ignited while the plane was parked in Boston and another forced an emergency landing in Japan when it began to smoke.

Since then, Boeing engineers have scrambled to insulate the eight cells in each battery, build a sturdier battery case and create a smoke-venting system to quell concerns about the battery’s safety and persuade regulators to lift the grounding order.