  A graph $G$ is \emph{uniquely k-colorable} if the chromatic number of $G$ is
$k$ and $G$ has only one $k$-coloring up to permutation of the colors. For a
plane graph $G$, two faces $f_1$ and $f_2$ of $G$ are \emph{adjacent
$(i,j)$-faces} if $d(f_1)=i$, $d(f_2)=j$ and $f_1$ and $f_2$ have a common
edge, where $d(f)$ is the degree of a face $f$. In this paper, we prove that
every uniquely 3-colorable plane graph has adjacent $(3,k)$-faces, where $k\leq
5$. The bound 5 for $k$ is best possible. Furthermore, we prove that there
exist a class of uniquely 3-colorable plane graphs having neither adjacent
$(3,i)$-faces nor adjacent $(3,j)$-faces, where $i,j\in \{3,4,5\}$ and $i \neq
j$. One of our constructions implies that there exist an infinite family of
edge-critical uniquely 3-colorable plane graphs with $n$ vertices and
$\frac{7}{3}n-\frac{14}{3}$ edges, where $n(\geq 11)$ is odd and $n\equiv
2\pmod{3}$.
