We consider multiple resonance scattering with complete frequency redistribution (CFR) in a semi-infinite conservative atmosphere (photon destruction probability ε₁ = 0) with the sources at infinite depth. The polarization arising in resonance scattering is completely accounted for. The problem we consider is the resonance-scattering counterpart of the Chandrasekhar-Sobolev problem of Rayleigh scattering in the conservative atmosphere. The numerical data on the matrix source function S(τ) in the atmosphere with conservative dipole resonance scattering (the depolarization parameter W = 1) are presented; we assume Doppler profile. The source matrix is found by a non-iterative numerical solution of the matrix Wiener-Hopf integral equation with the matrix Λ-operator. Depth dependence of the elements of the source matrix S(τ) is discussed. Some unexpected peculiarities are revealed in the behavior of its polarization terms. The matrix I(z) which is the generalization of the Chandrasekhar H-function to the case of polarized resonance scattering is found by the iterative solution of the Chandrasekhar-type nonlinear matrix integral equation. We present high-accuracy (5 s.f.) numerical data on I(z) for dipole conservative scattering with the Doppler profile. The center-to-limb variation of the degree of polarization in the core of a Doppler broadened resonance line is found. In conservative case, the limiting limb polarization δ₀ in the core of such a line is 9.4430% (for W = 1). The dependence of δ₀ on the depolarization parameter W is found. Simple interpolation formula, δ₀ = (9.443 - 38.05√ε₁)%, is suggested for the limb polarization of the radiation emerging from an isothermal nearly conservative atmosphere (ε₁ ≪ 1, W = 1). The data on I(z) are used to find the polarization line profiles and to trace their center-to-limb variation. The asymptotic expansions of S(τ) for τ → ∞ (deep layers) and of I(z) for z → ∞. (line wings) are found for the case of the Doppler profile. The coefficients of the expansions are determined by recursion relations. The numerical data on the accuracy and the domain of applicability of the asymptotic theory are presented.