In the last decade experimental and theoretical research has led to the discovery of double giant resonances—a giant resonance built on top of another giant resonance. The double-dipole resonance was first identified in a pion charge exchange reaction @1# and was predicted earlier @2#. Later the double dipole was detected in Coulomb excitation in heavyion reactions @3–5#. Properties of double dipole modes and other types of double resonances were studied and are reviewed in several review articles @2–4,6#. Theoretical studies have been mostly of the ‘‘macroscopic’’ type, introducing collective coordinates in the description of double giant resonances @2–4,6#. Some papers used semimicroscopic models in which the random phase approximation ~RPA! was employed to define the collective phonon states. Some extensions of the RPA have also been suggested @7,8#. Truly microscopic calculations are very difficult. A shell-model calculation requires very large spaces involving configurations made up of particles excited to several major shells. One must therefore truncate the space and limit the calculation to one-particle–one-hole (1p-1h) and two-particle– two-hole (2p-2h) configurations involving particles and holes in several major shells. Studies of this type were performed for some nuclei a few years ago @9#. In this work we present shell-model calculations of the double giant dipole state in O and Ca. By choosing light nuclei we are able to include a relatively large space of 1p-1h and 2p-2h configurations and therefore are able to study in detail the distribution of strength and the splitting of strength into the various allowed spin and isospin components. In the case of O we are able to account for the coupling of J50 and 2, 1p-1h states to the corresponding 2p-2h configurations. We also discuss energy weighted sum rules ~EWSR! and the relationship between these sum rules for the single and double giant resonances ~see also Ref. @10#!. The sum rules are evaluated in the shell model basis ~numerically! and in a boson model.