The Si-29(d, He-3)Al-28 reaction has been investigated at 29 MeV incident energy. Observations using a split-pole magnetic spectrograph have been made of 55 levels of Al-28 in the range of excitation energy between 0 and 6.7 MeV. Most of them have been identified with Al-28 levels which have been previously observed by other techniques. The spectroscopic factors have been obtained for 23 of these levels through distorted-wave Born approximation analyses of measured angular distributions. The levels at E(x)=3.105 and 3.762 MeV have been definitely assigned J(pi)=1+ and 0+, respectively. Four levels which are populated through the pickup of a l(p)=1 proton have been observed at E(x)=4.998, 5.406, 6.021, and 6.652 MeV. The excitation energies and spectroscopic factors for positive-parity states were compared with the results of a recent, complete sd-shell space, shell-model calculation. This comparison led to the identification of 21 shell-model levels with experimental levels. This comparison seems accurate enough to make very likely the J(pi)=3+ assignment for the levels at E(x)=2.988 and 4.597 MeV which were previously assigned J(pi)=(1,3)+.