Joubert, 28, was rushed to hospital after nurses at the bullring staunched the bleeding and was to be operated on overnight.
A bull raised in France gored him in the thigh, touching the femoral artery, but his life was not in danger.
The fights were stopped for half an hour to allow him to be cared for and taken to hospital.
Joubert was replacing another French bullfighter, Juan Leal, who was recently gored in the Spanish Basque city of Bilbao.
Bayonne is in the French Basque country, where bullfighting is still a popular sport, as it is elsewhere in the south-west, despite campaigns against it by animal-rights activists.
Last year an anti-bullfighting activist was gored after jumping into the ring, one of several such protests in recent years.