A measurement of the production cross section ratio sigma(chi[b2](1P)) / sigma( chi[b1](1P)) is presented. The chi[b1](1P) and chi[b2](1P) bottomonium states, promptly produced in pp collisions at sqrt(s) = 8 TeV, are detected by the CMS experiment at the CERN LHC through their radiative decays chi[b1,2](1P) to Upsilon(1S) + gamma. The emitted photons are measured through their conversion to electron-positron pairs, whose reconstruction allows the two states to be resolved. The Upsilon(1S) is measured through its decay to two muons. An event sample corresponding to an integrated luminosity of 20.7 inverse femtobarns is used to measure the cross section ratio in a phase-space region defined by the photon pseudorapidity, abs(eta[gamma]) < 1.0; the Upsilon(1S) rapidity, abs(y[Upsilon]) < 1.5; and the Upsilon(1S) transverse momentum, 7 < pt[Upsilon] < 40 GeV. The cross section ratio shows no significant dependence on the Upsilon(1S) transverse momentum, with a measured average value of 0.85 +/- 0.07 (stat+syst) +/- 0.08 (BF), where the first uncertainty is the combination of the experimental statistical and systematic uncertainties and the second is from the uncertainty in the ratio of the chi[b] branching fractions.