A dedicated study of the quenching of the weak axial-vector coupling strength g_A in nuclear processes has been performed by the COBRA collaboration. This investigation is driven by nuclear model calculations which show that the β-spectrum shape of the fourfold forbidden non-unique decay of ¹¹³Cd strongly depends on the effective value of g_A. Using an array of CdZnTe semiconductor detectors, 45 independent ¹¹³Cd spectra were obtained and interpreted in the context of three nuclear models. The resulting effective mean values are g‾_A(ISM)=0.915±0.007, g‾_A(MQPM)=0.911±0.013 and g‾_A(IBFM-2)=0.955±0.022. These values agree well within the determined uncertainties and deviate significantly from the free value of g_A. This can be seen as a first step towards answering the long-standing question regarding quenching effects related to g_A in low-energy nuclear processes.