We study L-p-mapping properties of the rough singular integral operator T(n
u)f(x) = integral (infinity)(0) dr/r integral (Sigma n-1) f(x-r theta )d nu
(theta )depending on a finite Borel measure nu on the unit sphere Sigma (n-
1) in R-n. It is shown that the conditions sup (\ xi \ = 1) integral (Sigma
n-1) log (1/\ theta . xi \ )d \v \(theta) < <infinity>, nu(Sigma (n-1)) =
0 imply the L-p-boundedness of T-nu for all 1 < p < infinity provided that
n >2 and nu is zonal.