High-precision measurements of the ft values for superallowed Fermi β decays between 0+ isobaric analogue states have, for decades, provided demanding tests of the Standard Model description of electroweak interactions. In order to significantly contribute to these tests experimentally, β decay half-lives and branching ratios must be determined to overall precisions of ± 0.05% or better, and β decay Q values must be deduced to at least ± 0.01%. For β decay half-lives in particular, this demanding requirement is generally accomplished using direct β counting techniques. This method was employed as part of this thesis in order to deduce the half-life of the superallowed β + emitter 62Ga using mass-separated radioactive ion beams provided by the Isotope Seperator and Accelerator (ISAC) facility at TRIUMF. The result of this analysis, T1/2(62Ga)β = 116.100 ± 0.025 ms, is now the single most precise superallowed half-life ever reported. In cases where there are large amounts of contaminant or daughter activities, one must instead rely on a measurement of the half-life using the γ-ray activity. Half-life measurements using the technique of γ-ray photopeak counting have, however, been previously limited by a systematic bias associated with detector pulse pile-up effects. While detector pulse pile-up has been qualitatively understood for decades, there has not been a quantitative description of its effects on half-life measurements to the level of precision required (± 0.05%) for superallowed Fermi β decay studies. Using the 8π γ-ray spectrometer, a spherical array of 20 HPGe detectors at ISAC, a new method was developed that, for the first time, provides the necessary quantitative description of detector pulse pile-up to the required level of precision. This novel technique has been verified through both a detailed Monte-Carlo simulation and experimentally using radioactive beams of 26Na. Following a correction of nearly 30 statistical standard deviations for pulse pile-up, the half-life of 26Na deduced in this work, T1/2(26Na)γ = 1.07167 ± 0.00055 s, is precise to the level of 0.05% and is in excellent agreement with the corresponding value, T1/2(26Na)β = 1.07128 ± 0.00025 s, deduced from direct β counting. This study has demonstrated the feasibility of using the γ-ray counting technique to deduce β decay half-lives to the necessary level of ± 0.05% precision. As an extension to this work, the half-life of the superallowed β + emitter 18Ne was determined to be, T1/2(18Ne)γ = 1.6656 ± 0.0019 s, a result that is a factor of four times more precise than the previous world average.