  We present an exact axially symmetric traversable wormhole of Einstein's
field equations in vacuum. This solution is a geometrically modified version of
the \textit{Yilmaz} \textit{exponential metric} which has recently been shown
to represent a traversable wormhole. This vacuum wormhole is a one-parameter
static singular solution with its singularity placed at $r=0$. The throat is
located at $r\simeq 0.5M$ in which $M$ is the total mass of the wormhole. In
analogy with the exponential wormhole, we show that this modified exponential
wormhole satisfies the flare-out conditions. Furthermore, it is shown that the
circular stable orbit doesn't exist for a null particle, however, for a massive
particle such exists provided the radius of the circular orbit is larger than a
minimum radius. This is in analogy with the Schwarzschild as well as the
exponential metric.
