Protons are the most abundant element in the galactic cosmic radiation, and the energy spectrum peaks around 1 GeV. Shielding of relativistic protons is therefore a key problem in the radiation protection strategy of crewmembers involved in long-term missions in deep space. Hydrogen ions were accelerated up to 1 GeV at the NASA Space Radiation Laboratory, Brookhaven National Laboratory, New York. The proton beam was also shielded with thick (about 20 g/cm2) blocks of lucite (PMMA) or aluminium (Al). We found that the dose rate was increased 40-60% by the shielding and decreased as a function of the distance along the axis. Simulations using the General-Purpose Particle and Heavy-Ion Transport code System (PHITS) show that the dose increase is mostly caused by secondary protons emitted by the target. The modified radiation field after the shield has been characterized for its biological effectiveness by measuring chromosomal aberrations in human peripheral blood lymphocytes exposed just behind the shield block, or to the direct beam, in the dose range 0.5-3 Gy. Notwithstanding the increased dose per incident proton, the fraction of aberrant cells at the same dose in the sample position was not significantly modified by the shield. The PHITS code simulations show that, albeit secondary protons are slower than incident nuclei, the LET spectrum is still contained in the low-LET range (<10 keV/microm), which explains the approximately unitary value measured for the relative biological effectiveness.