Boeing’s 787s could be flying again within weeks, a major milestone for the innovative passenger jets that have been grounded since January because of battery problems.

The Federal Aviation Administration on Friday approved Boeing’s plans to fix the plane’s lithium-ion batteries after two erupted in smoke and fire on separate planes. Boeing has deployed teams of technicians around the world to quickly install the modified batteries on the 50 jets that have been delivered so far and return them to service as soon as possible.

More important, the F.A.A.’s decision frees Boeing to resume deliveries of 787s in the next two months, a critical step for the plane maker and for airlines that have been eagerly awaiting the new, more fuel-efficient jets. Boeing said Friday that it would deliver all 787s that were planned this year.

The battery problems have probably cost Boeing hundreds of millions of dollars, and airlines are likely to seek financial compensation for the delays. Still, Boeing has not seen an impact on the 800 orders it has booked for the plane, which promises fuel cost savings of 20 percent. The 787 is the first commercial aircraft built largely from lighter carbon-composite materials, and it uses more electrical systems than conventional airplanes.