We report, to the best of our knowledge, the first experimental observation of coexistence of scalar dark and bright solitons in a fiber laser operating in the near zero-dispersion regime. We show that because of the incoherent cross coupling, under suitable conditions a bright soliton formed in the net anomalous cavity dispersion regime could bind with a dark soliton formed in the net normal cavity dispersion regime in a fiber laser. The properties of the dark and bright solitons, as well as their bound states, are experimentally investigated. The numerical simulations based on the coupled Ginzburg–Landau equations have reproduced the experimental results well.