The decays B-c(+) -> J/psi D-s(+) and B-c(+) -> J/psi D-s*(+) are studied with the ATLAS detector at the LHC using a dataset corresponding to integrated luminosities of 4.9 and 20.6 fb(-1) of pp collisions collected at centre-of-mass energies root s = 7 TeV and 8 TeV, respectively. Signal candidates are identified through J/psi -> mu(+)mu(-) and D-s(()*()+) -> phi pi(+)(gamma/pi(0)) decays. With a two-dimensional likelihood fit involving the B-c(+) reconstructed invariant mass and an angle between the mu(+) and D-s(+) candidate momenta in the muon pair rest frame, the yields of B-c(+) -> J/psi D-s(+) and B-c(+) -> J/psi D-s*(+), and the transverse polarisation fraction in B-c(+) -> J/psi D-s*(+) decay are measured. The transverse polarisation fraction is determined to be Gamma +/-+/-(B-c(+) -> J/psi D-s*(+))/Gamma(B-c(+) -> J/psi D-s*(+)) = 0.38 +/- 0.23 +/- 0.07, and the derived ratio of the branching fractions of the two modes is B-Bc+ -> J/psi D-s*+/B-Bc+ -> J/psi D-s(+) = 2.8(-0.8)(+1.2) +/- 0.3, where the first error is statistical and the second is systematic. Finally, a sample of B-c(+) -> J/psi pi(+) decays is used to derive the ratios of branching fractions B-Bc+ -> J/psi D-s*+/B-Bc+ -> J/psi pi(+) = 3.8 +/- 1.1 +/- 0.4 +/- 0.2 and B-Bc+ -> J/psi D-s*+/B-Bc+ -> J/psi pi(+) = 10.4 +/- 3.1 +/- 1.5 +/- 0.6, where the third error corresponds to the uncertainty of the branching fraction of D-s(+) -> phi(K+ K-)pi(+) decay. The available theoretical predictions are generally consistent with the measurement.