[en] Electron capture Β-decay at low temperatures is discussed. It is shown that the capture of s-electrons, from inner closed electronic shells in many-electron atoms and ions, is considerably altered by the hyperfine interaction. The capture rate can depend significantly on the hyperfine state of the decaying isotope. The total decay rate of a sample of a capture isotope will therefore depend on the populations of the hyperfine states and hence on temperature. The effect is shown to persist in the presence of a magnetic field. The effects should be observable experimentally in paramagnetic atoms or ions cooled to temperatures comparable to the hyperfine splitting. These conditions can be achieved using laser cooling in the gas phase or with conventional cryogenic techniques in solids. Potential applications in neutrino research and nuclear physics are described