We　propose　a　scheme　for　implementation　of　a　universal　set　of　quantum　logic　gates　in　decoherence-free　subspace　with　atoms　trapped　in　distant　cavities　connected　by　optical　fibers.　The　selective　dispersive　couplings　between　the　ground　states　and　the　first-excited　states　of　the　atom-cavity-fiber　system　produce　a　state-dependent　Stark　shift,　which　can　be　used　to　implement　nonlocal　phase　gates　between　two　logic　qubits.　The　single-logic-qubit　quantum　gates　are　achieved　by　the　local　two-atom　collision　and　the　Stark　shift　of　a　single　atom.　During　all　the　logic　operations,　the　logic　qubits　remain　in　decoherence-free　subspace　and　thus　the　operation　is　immune　to　collective　dephasing.