We present a study of B0s decays to the CP-odd final state J/ψf0(980) with J/ψ→μ+μ− and f0(980)→π+π−. Using pp collision data with an integrated luminosity of 3.8  fb−1 collected by the CDF II detector at the Tevatron we measure a B0s lifetime of τ(B0s→J/ψf0(980))=1.70+0.12−0.11(stat)±0.03(syst)ps. This is the first measurement of the B0s lifetime in a decay to a CP eigenstate and corresponds in the standard model to the lifetime of the heavy B0s eigenstate. We also measure the product of branching fractions of B0s→J/ψf0(980) and f0(980)→π+π− relative to the product of branching fractions of B0s→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.