Polymer micelles represent one of the most attractive drug delivery systems due to their design flexibility based on a variety of macromolecular synthetic methods. The environmentally safe chemistry in which the use or generation of hazardous materials is minimized has an increasing impact on polymer-based drug delivery nanosystems. In this work, a solvent-free green synthetic procedure was applied for the preparation of an amphiphilic diblock copolymer consisting of biodegradable hydrophobic poly(acetylene-functional carbonate) and biocompatible hydrophilic polyethylene glycol (PEG) blocks. The cyclic functional carbonate monomer 5-methyl-5-propargyloxycarbonyl-1,3-dioxane-2-one (MPC) was polymerized in bulk using methoxy PEG-5K as a macroinitiator by applying the metal-free organocatalyzed controlled ring-opening polymerization at a relatively low temperature of 60 °C. The functional amphiphilic block copolymer self-associated in aqueous media into stable micelles with an average diameter of 44 nm. The copolymer micelles were physico-chemically characterized and loaded with the plant-derived anticancer drug curcumin. Preliminary in vitro evaluations indicate that the functional copolymer micelles are non-toxic and promising candidates for further investigation as nanocarriers for biomedical applications.