Human infection with avian influenza A(H7N9) virus is associated mainly with the exposure to infected poultry. The factors that allow interspecies transmission but limit human-to-human transmission are unknown. Here we show that A/Anhui/1/2013(H7N9) influenza virus infection of chickens (natural hosts) is asymptomatic and that it generates a high genetic diversity. In contrast, diversity is tightly restricted in infected ferrets, limiting further adaptation to a fully transmissible form. Airborne transmission in ferrets is accompanied by the mutations in PB1, NP and NA genes that reduce viral polymerase and neuraminidase activity. Therefore, while A(H7N9) virus can infect mammals, further adaptation appears to incur a fitness cost. Our results reveal that a tight genetic bottleneck during avian-to-mammalian transmission is a limiting factor in A(H7N9) influenza virus adaptation to mammals. This previously unrecognized biological mechanism limiting species jumps provides a measure of adaptive potential and may serve as a risk assessment tool for pandemic preparedness.