The decoration of carbon nanotubes (CNTs) by metal nanoparticles (NPs) combines the advantages of a high specific surface material with catalytic properties of metal nanocrystals. Little work has been devoted to the decoration of CNTs with copper NPs, and no evidence of copper atomic decoration of CNTs has shown up until now. Herein, we demonstrate that the strong acidic oxidation of double-walled CNTs (dwCNTs) is very efficient for the decoration of the carbon surface by copper NPs and atoms. This treatment severely degraded the CNT walls and generated a large amount of disordered sp3 carbon. This amorphous carbon film bears many chemically active functions like carboxyl and hydroxyl ones. In such conditions, the CNT walls behave as very efficient ligands for the stabilization of copper obtained by the thermolysis of the mesityl precursor in organic solution under mild dihydrogen pressure. In addition to copper NPs, we evidenced the presence of a regular coverage with copper atoms over the dwCNTs. This nanocomposite catalyzes the quantitative synthesis of propargylamines via one A3-type coupling reaction. Five consecutive catalytic cycles with 100% yield could be performed with no loss of activity, and the combination of Cu supported on dwCNTs allows a facile recycling of the catalytic material.