The judge halted the trial for several months because several of the defendants who reside in France were not present, Radio France Internationale said.
The defendants include a man who allegedly posed as an Algerian doctor and ran a clinic where unmarried Algerian women were paid to give up their babies for adoption in France.
RFI said some of the women claimed they were threatened and forced to give up their children.
The trafficking ring was broken up after a woman died while undergoing an abortion in 2009.
Caroline Berg Eriksen: Soccer player's wife triggers debate with post-birth selfie
Texas principal bans speaking Spanish, stirs controversy