The paper deals with Jacobi matrices with weights λ_k given by λ_k=k^α(1+Δ_k), where α∈(12, 1) and lim_kΔ_k=0. The main question studied here concerns when the spectrum of the operator J defined by the Jacobi matrix has absolutely continuous component covering the real line. A sufficient condition is given for a positive answer to the above question. The method used in the paper is based on a detailed analysis of generalized eigenvectors of J. In turn this analysis relies on the so-called grouping in blocks approach to a large product of the transfer matrices associated to J.