We have performed elastic- and inelastic-neutron-scattering experiments on a single crystal of the quasi-one-dimensional system CuGe_0.993Si_0.007O₃. Our experimental data show unambiguously that both quasi-gapless antiferromagnetic fluctuations and gapped magnetic fluctuations, associated with a dimerized phase, coexist in this material below a spin-Peierls transition temperature T_SP ≈ 9.2 K. When the temperature is lowered below T_N ≈ 4 K, part of the system undergoes a new phase transition towards long-range antiferromagnetism, coexisting with the dimerized non-magnetic phase. We will describe the temperature and wave vector dependences of magnetic responses in these two phases, which can be understood as the superposition of two independent spectra.