We consider a two-dimensional quantum waveguide composed of two semi-strips of width 1and 1 − ε, where ε > 0 is a small real parameter,i.e. the waveguide is gently converging. The width of the junction zonefor the semi-strips is 1 + O(√ε). We will present a sufficient condition for the existence of a weaklycoupled bound state below π2, the lower bound of thecontinuous spectrum. This eigenvalue in the discrete spectrum is unique and itsasymptotics is constructed and justified whenε → 0+.