We analyze an evolving network model of Krapivsky and Redner in which new nodes
arrive sequentially, each connecting to a previously existing node $b$ with
probability proportional to the $p$th power of the in-degree of $b$. We
restrict to the super-linear case $p>1$. When $1+\frac{1}{k} < p < 1 +
\frac{1}{k-1}$, the structure of the final countable tree is determined. There
is a finite tree $\mbox{T}$ with distinguished $v$ (which has a limiting
distribution) on which is ``glued" a specific infinite tree; $v$ has an
infinite number of children, an infinite number of which have $k-1$ children,
and there are only a finite number of nodes (possibly only $v$) with $k$ or
more children. Our basic technique is to embed the discrete process in a
continuous time process using exponential random variables, a technique that
has previously been employed in the study of balls-in-bins processes with
feedback.