The CP violating phase β^J/ψφ_s is measured in decays of B⁰_s → J/ψφ. This measurement uses 5.2 fb^-1 of data collected in √s = 1.96 TeV p‾p collisions at the Fermilab Tevatron with the CDF Run-II detector. CP violation in the B⁰_s-‾B⁰_sbar system is predicted to be very small in the Standard Model. However, several theories beyond the Standard Model allow enhancements to this quantity by heavier, New Physics particles entering second order weak mixing box diagrams. Previous measurements have hinted at a deviation from the Standard Model expectation value for β^J/ψφ_s with a significance of approximately 2σ. The measurement described in this thesis uses the highest statistics sample available to date in the B⁰_s → J/ψφ decay channel, where J/ψ → μ⁺ μ^- and φ → K⁺K^-. Furthermore, it contains several improvements over previous analyses, such as enhanced signal selection, fully calibrated particle ID and flavour tagging, and the inclusion of an additional decay component in the likelihood function. The added decay component considers S-wave states of KK pairs in the B⁰_s → J/ψK⁺K^- channel. The results are presented as 2-dimensional frequentist confidence regions for β^J/ψφ_s and ΔΓ (the width difference between the B⁰_s mass eigenstates), and as a confidence interval for β^J/ψφ_s of [0.02,0.52] U [1.08, 1.55] at the 68% confidence level. The measurement of the CP violating phase obtained in this thesis is complemented by the world's most precise measurement of the lifetime τ_s = 1.53 ± 0.025 (stat.) ± 0.012 (syst.) ps and decay width difference ΔΓ = 0.075 ± 0 .035 (stat.) ± 0.01 (syst.) ps⁻¹ of the B⁰_s meson, with the assumption of no CP violation.