We propose a scheme to prepare the Bell states for atomic qubits trapped in separate optical cavities via atom-cavity-laser interaction. The quantum information of each qubit is encoded on the degenerate ground states of the atom, so the entanglement between them is relatively stable against spontaneous emission. The proposed scheme consists of a Mach-Zehnder interferometer (MZI) with two arms, and each arm contains a cavity with an N-type atom in it. It requires two classical fields and a single-photon source. By controlling the sequence and time of atom-cavity-laser interaction, the deterministic production of the atomic Bell states is shown. We also introduce the generalization of the present scheme to generate the 2N-atom Greenberger-Horne-Zeilinger state.