We took Cala on an airplane when she was 3 weeks, 1 day old. We adopted her from Wyoming and we live in Alabama. She did not have to have medical clearance to fly. But we did ask the doctor who had delivered her. He had actually seen her for two appointments while we were waiting for clearance to come home anyway. Some airlines do require a letter of clearance, so ask your airline, and talk to your pediatrician.
We brought Cala's baby bucket (carrier) on the airplane, and she pretty much stayed in there for the whole flight. I think I did get her out for just a few minutes of the two flights. I also timed it so that she was getting a bottle during take-offs and landings. Since Robert had to go back to work my Mom had come out to stay with us while we waited to go home. For the first flight my Mom was sitting with me, but for the second Mom & I were separated. We all did great on the plane. Everyone commented how Cala was so quiet and good on both of our flights.
We were very fortunate to have an empty seat beside us on each flight, so I could strap the bucket in with the seatbelt.
We had not brought our stroller with us, because we just didn't see how we could have carried it, and the luggage. The bucket pops into the stroller, so we might have been able to do it? Who knows? Once we were up there we REALLY wished that we had brought the stroller. You may want to try to bring yours along. The flight attendants will probably take it and stow it for you, and then get it out for you as you de-plane.
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if you have any other questions. I will try to help. Cala was/is such a good baby/little girl that she made our travel really easy. I did get a lot of strange looks though travelling with such a tiny baby.