A method is disclosed for forming copper damascene interconnects without the attendant CMP (chemical-mechanical polishing) dishing problem that is encountered in the art. This is accomplished by first lining the inside walls of a dual damascene structure with a diffusion barrier layer, and then depositing copper metal into the damascene structure. Secondly, as a key aspect of the invention, and before removing the excess copper either by conventional etching techniques-which is difficult for copper- or by conventional CMP- which causes dishing in grooves or trenches- an etch stop layer is deposited covering the copper layer. Portions of the etch stop layer is next removed from the high regions of the underlying copper by a quick first CMP so that other portions of the etch stop layer over the wider trenches/groove remain low and unaffected. The high regions now exposed are etched while the low regions protected by the etch-stop layer still remain unaffected. When the copper etching reaches a level below the level of the low regions, a global CMP is performed so that all of the excess copper is removed and the level of the copper especially over the wider trench areas reaches within very close proximity of the level of the insulation layer surrounding the copper damascene- and without dishing.