  We introduce and study the dynamics of an \emph{immortal} critical branching
process. In the classic, critical branching process, particles give birth to a
single offspring or die at the same rates. Even though the average population
is constant in time, the ultimate fate of the population is extinction. We
augment this branching process with immortality by positing that either: (a) a
single particle cannot die, or (b) there exists an immortal stem cell that
gives birth to ordinary cells that can subsequently undergo critical branching.
We discuss the new dynamical aspects of this immortal branching process.
