A frozen laser target that is in the gaseous phase at room temperature is advantageous for a laser ion source because it can avoid the accumulation of damage from laser irradiation by regeneration of the target surface by additional gas freezing. In this study, the possibility of forming a xenon ion beam with a current sufficient for heavy-ion inertial fusion (HIF) was investigated. The relationship between the frozen target growth and resulting ion current was analyzed to determine the optimal condition of laser irradiation that ensures stable supply of ion beams for a long time. A frozen target of xenon was formed on a mount cooled to 20 K using a Gifford McMahon cryocooler, and plasma was generated using a Nd:YAG laser. The results showed that it is possible to supply a sufficient ion current for application of singly charged ions as a driver for HIF. Additionally, it was indicated that the xenon responsible for forming the plasma existed only at a certain depth from the target surface. The results imply that it is possible to obtain a stable ion supply for a long time by irradiating the target with a laser after the frozen xenon grows to a sufficient thickness.