A helicon mode discharge is realized in a newly erected set-up and two jumps are observed in the variation of the plasma density with the controlled magnetic field, which is a typical characteristic of the helicon mode discharge. A reasonable explanation is proposed for the appearance of the density jumps. The helicon wavelength is evaluated to be about 20 cm and it is nearly constant in the helicon mode discharge. The density obtained is nearly up to 3.5×1018m-3 with only 0.5 Pa argon pressure and 300 W radio frequency power supply, and it is more than one magnitude higher than that in other discharges at comparable pressures and input powers. The electron temperature is found to increase as the pressure is decreased.