We present a study of Bs0 decays to the CP-odd final state J/ψf0(980) with J/ψ→μ+μ- and f0(980)→π+π-. Using pp̅ collision data with an integrated luminosity of 3.8  fb-1 collected by the CDF II detector at the Tevatron we measure a Bs0 lifetime of τ(Bs0→J/ψf0(980))=1.70-0.11+0.12(stat)±0.03(syst)ps. This is the first measurement of the Bs0 lifetime in a decay to a CP eigenstate and corresponds in the standard model to the lifetime of the heavy Bs0 eigenstate. We also measure the product of branching fractions of Bs0→J/ψf0(980) and f0(980)→π+π- relative to the product of branching fractions of Bs0→J/ψϕ and ϕ→K+K- to be Rf0/ϕ=0.257±0.020(stat)±0.014(syst), which is the most precise determination of this quantity to date.