In this work weakly essential g-supplemented modules are definedand some properties of these modules are investigated. All modules areassociative with unity and all modules are unital left modules. Let N bean R-module. If every essential submodule of N has a weak g-supplementin M, then M is called a weakly essential g-supplemented (or briefly wegsupplemented) module. Clearly we can see that every weakly gsupplemented module is weakly essential g-supplemented.Every weakly essential supplemented module is also weaklyessential g-supplemented. Because of this weakly essential gsupplemented modules are more generalized than weakly essentialsupplemented modules. Every (generalized) hollow and every localmodule are weakly essential g-supplemented. Let N be a weakly essentialg-supplemented R-module. If every nonzero submodule of N is essentialin N, then N is weakly g-supplemented. It is showed that every factormodule and every homomorphic image of a weakly essential gsupplemented module are weakly essential g-supplemented. It is alsoproved that the finite sum of weakly essential g-supplemented modulesis weakly essential g-supplemented. Let N be a weakly essential gsupplemented R-module. Then N/RadgN have no proper essentialsubmodules. Let N be a weakly essential g-supplemented R-module. Thenevery finitely N-generated R-module is weakly essential g-supplemented.Let R be any ring. Then RR is weakly essential g-supplemented if and onlyif every finitely generated R-module is weakly essential g-supplemented.Let N be an R-module. If every essential submodule of N is g* equivalentto an weak g-supplement submodule in N, then N is weakly essential gsupplemented.