Final answer:
The process described for a single-celled organism to produce offspring is called asexual reproduction through mitosis, resulting in genetically identical daughter cells.
Step-by-step explanation:
The process described where a single-celled organism produces offspring is known as asexual reproduction, specifically through a method called mitosis. In this process, the organism duplicates its DNA and then divides into two separate cells. Each resulting daughter cell is genetically identical to the original parent cell, thereby continuing the species. This form of reproduction is efficient for single-celled organisms and allows for rapid population growth under favorable conditions.