The MoEDAL experiment is designed to search for magnetic monopoles and other highly-ionising particles produced in high-energy collisions at the LHC. The largely passive MoEDAL detector, deployed at Interaction Point 8 on the LHC ring, relies on two dedicated direct detection techniques. The first technique is based on stacks of nuclear-track detectors with surface area similar to 18 m(2), sensitive to particle ionisation exceeding a high threshold. These detectors are analysed offline by optical scanning microscopes. The second technique is based on the trapping of charged particles in an array of roughly 800 kg of aluminium samples. These samples are monitored offline for the presence of trapped magnetic charge at a remote superconducting magnetometer facility. We present here the results of a search for magnetic monopoles using a 160 kg prototype MoEDAL trapping detector exposed to 8TeV proton-proton collisions at the LHC, for an integrated luminosity of 0.75 fb(-1). No magnetic charge exceeding 0.5g(D) (where g(D) is the Dirac magnetic charge) is measured in any of the exposed samples, allowing limits to be placed on monopole production in the mass range 100 GeV <= m <= 3500 GeV. Model-independent cross-section limits are presented in fiducial regions of monopole energy and direction for 1g(D) <= vertical bar g vertical bar <= 6g(D), and model-dependent cross-section limits are obtained for Drell-Yan pair production of spin-1/2 and spin-0 monopoles for 1g(D) <= vertical bar g vertical bar <= 4g(D). Under the assumption of Drell-Yan cross sections, mass limits are derived for vertical bar g vertical bar = 2g(D) and vertical bar g vertical bar = 3g(D) for the first time at the LHC, surpassing the results from previous collider experiments.