Let $G$ be a connected graph of order $n$. The eccentricity $e(v)$ of a vertex $v$ is the distance from $v$ to a vertex farthest from $v$. The average eccentricity of $G$ is the mean of all eccentricities in $G$. We give upper bounds on the average eccentricity of $G$ in terms of order $n$, minimum degree $δ$, and girth $g$. In addition, we construct graphs to show that, if for given $g$ and $δ$, there exists a Moore graph of minimum degree $δ$ and girth $g$, then the bounds are asymptotically sharp. Moreover, we show that the bounds can be improved for a graph of large degree $Δ$.