  We consider a variation on the Barab\'asi-Albert random graph process with
fixed parameters $m\in \mathbb{N}$ and $1/2 < p < 1$. With probability $p$ a
vertex is added along with $m$ edges, randomly chosen proportional to vertex
degrees. With probability $1 - p$, the oldest vertex still holding its original
$m$ edges loses those edges. It is shown that the degree of any vertex either
is zero or follows a geometric distribution. If $p$ is above a certain
threshold, this leads to a power law for the degree sequence, while a smaller
$p$ gives exponential tails. It is also shown that the graph contains a unique
giant component whp if and only if $m\geq 2$.
