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Planar MgB2/TiB2/MgB2 junctions were fabricated by locally removing MgB2 within a gap of les 50 nm on a bilayer film of MgB2/TiB2 on SiC substrate using focused ion beam (FIB). Junctions with gaps of different depths were studied. We found that when the gap is too shallow the junction shows a flux-flow behavior, while a too deep gap will result in a junction with no supercurrent. Only junctions with the right gap depth exhibit good resistively-shunted-junction (RSJ) like current-voltage characteristics and excellent ac Josephson effect. The junctions' behavior can be described by the proximity effect superconductor-normal metal-superconductor (SNS) model, suggesting that the Josephson coupling is through the proximity effect in the metallic TiB2 layer.