This is the first time that such a bill clears this first parliamentary hurdle, according to Soledad Luque of Coordinadora X24, which groups victims' associations in Spain.
“It's the first time in all the years that we've been fighting,” she told AFP. “It's historic.”
The bill has to clear several stages before it can be voted into law.
Among the measures it envisages is a nationwide database of all people who may have been affected.
It also wants to create a DNA bank of potential victims, and cross-check this with DNA from exhumed remains.
Scores of babies were taken from their mothers — who were told their children had died — and given to others to adopt during and after Franco's 1939-1975 dictatorial rule.
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Estimates range from hundreds to tens of thousands of victims.
Doctors played a major part in the scheme to provide infertile couples — preferably those close to the regime — with stolen newborns, often with the help of the Catholic Church.
Initially, babies were taken from left-wing opponents of the regime, with the practice later expanded to supposedly illegitimate children and those from poor families.
The newborns were meant to be raised by affluent, conservative and devout Roman Catholic families.
Even after Spain transitioned to democracy following Franco's death in 1975, the trafficking went on until at least 1987.
In October, an elderly Spanish doctor was found guilty of taking a newborn baby from her mother for illegal adoption in 1969 in a landmark trial.
He escaped punishment, however, because the court decided Ines Madrigal, the “stolen baby” who is now 49, waited too long to file a complaint.
ANALYSIS: The 'stolen babies' trial in Spain finally shines a light on …