Minimum diameter spanning tree problem
The answer is given in the paper that you link to. Specifically, there is a $O(mn+n^2 \log n)$-time algorithm for the problem (with positive edge weights, as required) that works as follows.

The absolute 1-center is the point in the graph - either a vertex or an edge - from which the distance to the furthest vertex is minimum. Now, denote by $T(v)$ a shortest path tree connecting $v$ to all vertices in $V$. It holds that $T(a)$ is a minimum diameter spanning tree of $G$, where $a$ is an absolute 1-center of $G$. Here, if the absolute 1-center is an edge, we subdivide it to obtain the vertex $a$ from which $T(a)$ is then computed from.

An absolute 1-center $a$ can be found by the Kariv-Hakimi algorithm within the mentioned bound, while $T(a)$ is computed by any standard shortest path algorithm.