Aiming at synthesis of nonperiodic sequence-controlled polymers, we have been focusing on copper(I)-catalyzed azide-alkyne cycloaddition (CuAAC) polymerization of monomers possessing both azide and alkyne groups. For this purpose, we have chosen N-ethynyl-4-azidobenzenesulfonamide derivatives because the derivatives carrying various substituents can be prepared in a fewer steps using the corresponding starting materials, i.e., amines. In this study, we synthesized a new monomer, i.e., N-butyl-N-ethynyl-4-azidobenzenesulfonamide (BuEA), and carried out its dimerization and polymerization by CuAAC using a pair of copper(II) sulfate pentahydrate and sodium ascorbate. A dimer of BuEA was successfully obtained from the monomer precursors possessing azide and ethynyl groups, respectively. This observation indicates that N-ethynyl-4-azidobenzenesulfonamide derivatives are applicable to synthesis nonperiodic sequence-controlled polymers. The polymer obtained (poly (BuEA)) was well soluble in common organic solvents, e.g., dimethyl sulfoxide, N,N-dimethylformamide, tetrahydrofuran (THF), chloroform, and dichloromethane. Solution properties of poly(BuEA) were also investigated. Pulse-field-gradient spin-echo (PGSE) NMR data in THF-d(8) indicated that poly(BuEA) chain takes a compact conformation. Steady state fluorescence study exhibited that poly(BuEA) emitted weak fluorescence based on the monomer unit.