Electron spin resonance measurements were performed on a single crystal of the one-dimensional antiferromagnet CuGeO3 which shows the spin-Peierls transition. The signals in the paramagnetic phase show a strong interchain exchange interaction and the angular dependence of the g-value is explained by a tetragonal field. Decrease of absorption intensity dire to formation of the energy gap below the transition temperature is found. The angular-dependent linewidth below the transition temperature shows a different pattern from that of the paramagnetic phase. The temperature and angular dependence of the linewidth are discussed by taking account of the one-dimensional diffusion process and spin-Peierls system.