The aim of this study was to synthesize an intrinsically stretchable conductive polymer (CP) by atom transfer radical polymerization (ATRP). For this purpose, poly(3,4-ethyilenedioxythiophene) (PEDOT) was synthesized as a backbone, while poly(acrylate-urethane) (PAU) was grafted onto the PEDOT backbone to form graft polymers PEDOT-g-PAU. Different concentrations of acrylate-urethane (AU) were used to synthesize PAU side chains of different lengths. The successful synthesis of the obtained intermediates and products (PEDOT-g-PAU) was confirmed by infrared spectroscopy and nuclear magnetic resonance. Thermal properties were evaluated by differential scanning calorimetry and thermogravimetric analysis, while conductivity was determined by four-point probe measurement. A simple tensile test was performed to characterize the ductility of the samples. PEDOT-g-PAU has shown high stretchability of up to 500% and, therefore, could potentially be used in skin-worn flexible electronics, while additional subsequent doping is required to improve the deterioration of electrical properties after the addition of the insulating urethane layer.