Entire functions with preassigned zero proximate order

It is known that if the proximate order $\rho(r)$ is 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 an entire function has remained open until now. This question is investigated in the paper.