We consider the Hilbert-type operator defined byHÏ(f)(z)=â«01f(t)(1zâ«0zBtÏ(u)du)Ï(t)dt, where {BζÏ}ζâD are the reproducing kernels of the Bergman space AÏ2 induced by a radial weight Ï in the unit disc D. We prove that HÏ is bounded from Hâ to the Bloch space if and only if Ï belongs to the class DË, which consists of radial weights Ï satisfying the doubling condition sup0â©œr<1âĄâ«r1Ï(s)dsâ«1+r21Ï(s)ds<â. Further, we describe the weights ÏâDË such that HÏ is bounded on the Hardy space H1, and we show that for any ÏâDË and pâ(1,â), HÏ:L[0,1)pâHp is bounded if and only if the Muckenhoupt type conditionsup0<r<1âĄ(1+â«0r1ÏË(t)pdt)1p(â«r1Ï(t)pâČdt)1pâČ<â, holds. Moreover, we address the analogous question about the action of HÏ on weighted Bergman spaces AÎœp.