The as-yet-unnamed material - a form of artificial rubber - is made from vegetable oil and a component of urine.
The substance, described in the journal Nature, produces surfaces when cut that retain a strong chemical attraction to each other.
Pieces of the material join together again as if never parted without the need for glue or a special treatment.
This remarkable property comes from careful engineering of the molecules in the material.
The French researchers are already making kilogramme quantities in their Paris laboratories and say the process is almost completely green, and could be completely so with a few adjustments.
The secret of the substance lies in how the molecules are held together.
A piece of normal rubber, says Dr Ludwik Leibler, who headed the research, is actually a single molecule with billion upon billions of smaller units chemically welded together to form a giant tangled network.
But break a rubber (or most other solids), and the chemical welds - known as covalent bonds - are also broken.
These cannot be remade. Nor can a piece of rubber be remoulded or reshaped.
"We wanted to see if we could make a rubber-like material using small molecules," Dr Leibler of the Industrial Physics and Chemistry Higher Educational Institution (ESPCI) in Paris told the BBC's Science In Action programme.
The trick was to replace the covalent bonds in rubber with weaker connections known as hydrogen bonds.
These are like hands on neighbouring molecules that can clasp together, but let go when broken.
Dr Leibler quickly realised that this meant not only that the new rubber could be recycled and remoulded many times over, but that if separated by a cut or break, the chemical hands at the fresh surfaces would still be waving about ready to bind again.
François Tournilhac, who runs Dr Leibler's laboratories, demonstrated the healing to me.
Using a razor blade he severed a thin strand of the yellowish material (the colour of corn oil), showed me the clean square faces, and then pressed them together.
Almost immediately, the grip was strong enough for him to hold the sample just at one end.
Within an hour the bonds had rebuilt themselves so thoroughly that it was possible to stretch the strand to twice its length without any sign of weakness where the cut had been made.
One obvious use, says Dr Leibler, is for self-healing seals.
Puncture a seal in a compression joint with a nail, and the hole would automatically repair itself.
He also has more playful suggestions.
"Why not use it to make children's toys? Children are always breaking their toys. Wouldn't it be nice if you could put them back together so easily?"
The material was developed with the support of the French company Arkema, which is already investigating whether it can be turned into a commercial product.