  It is known that if the proximate order $\rho(r)$ such that $\lim \rho(r) =
\rho > 0 (r \to \infty)$, then there exists an entire function $f(z)$ of
proximate order $\rho(r)$. In the case where $\rho = 0$ the question about the
existence of such entire function remained open until now. This question is
investigated in the paper.
