Police said they were studying video footage of the woman in the hospital in the eastern city of Nancy from where the boy named Lucas was taken late Tuesday but said they had no leads on the abductor's identity.
Authorities launched an appeal on radio, television and on noticeboards in train stations as well as motorways to try to track down the woman described as slim, of European origin, and aged between 16 and 20.
She reportedly walked into the maternity ward and made off with the infant, who was wearing blue pyjamas, as his 24-year-old mother slept. Lucas is her second child.
Police brought sniffer dogs to the hospital to try to find traces of the stolen child, as other officers studied video footage from cameras in streets and public transport in Nancy.
Police said they had received several calls from the public but no major line of inquiry was being pursued so far.