  Let $\L (f) = K[x][y; f\frac{d}{dx} ]$ be an Ore extension of a polynomial
algebra $K[x]$ over a field $K$ of characteristic zero where $f\in K[x]$. For a
given polynomial $f$, the automorphism group of the algebra $\L (f) $ is
explicitly described. The polynomial case $\L (0) = K[x,y]$ and the case of the
Weyl algebra $A_1= K[x][y; \frac{d}{dx} ]$ were done done by Jung (1942) and
van der Kulk (1953), and Dixmier (1968), respectively. In 1997, Alev and Dumas
proved that the algebras $\L (f)$ and $\L (g)$ are isomorphic iff $g(x) = \l
f(\alpha x+\beta )$ for some $\l, \alpha \in K\backslash \{ 0\}$ and $\beta\in
K$. In 2015, Benkart, Lopes and Ondrus gave a complete description of the set
of automorphism groups of algebras $\L(f)$. In this paper we complete the
picture, i.e. {\em given} the polynomial $f$ we have the explicit description
of the automorphism group of $\L (f)$.
