  We give a $(1+\epsilon)$-approximate distance oracle with $O(1)$ query time
for an undirected planar graph $G$ with $n$ vertices and non-negative edge
lengths. For $\epsilon>0$ and any two vertices $u$ and $v$ in $G$, our oracle
gives a distance $\tilde{d}(u,v)$ with stretch $(1+\epsilon)$ in $O(1)$ time.
The oracle has size $O(n\log n ((\log n)/\epsilon+f(\epsilon)))$ and
pre-processing time $O(n\log n((\log^3 n)/\epsilon^2+f(\epsilon)))$, where
$f(\epsilon)=2^{O(1/\epsilon)}$. This is the first $(1+\epsilon)$-approximate
distance oracle with $O(1)$ query time independent of $\epsilon$ and the size
and pre-processing time nearly linear in $n$, and improves the query time
$O(1/\epsilon)$ of previous $(1+\epsilon)$-approximate distance oracle with
size nearly linear in $n$.
