  We describe experiments with an optical frequency standard based on a laser
cooled $^{171}$Yb$^+$ ion confined in a radiofrequency Paul trap. The
electric-quadrupole transition from the $^2S_{1/2}(F=0)$ ground state to the
$^2D_{3/2}(F=2)$ state at the wavelength of 436 nm is used as the reference
transition. In order to compare two $^{171}$Yb$^+$ standards, separate
frequency servo systems are employed to stabilize two probe laser frequencies
to the reference transition line centers of two independently stored ions. The
experimental results indicate a relative instability (Allan standard deviation)
of the optical frequency difference between the two systems of $\sigma_y(1000
{\rm s})=5\cdot 10^{-16}$ only, so that shifts in the sub-hertz range can be
resolved. Shifts of several hertz are observed if a stationary electric field
gradient is superimposed on the radiofrequency trap field. The absolute optical
transition frequency of Yb$^+$ at 688 THz was measured with a cesium atomic
clock at two times separated by 2.8 years. A temporal variation of this
frequency can be excluded within a $1\sigma$ relative uncertainty of $4.4\cdot
10^{-15}$ yr$^{-1}$. Combined with recently published values for the constancy
of other transition frequencies this measurement provides a limit on the
present variability of the fine structure constant $\alpha$ at the level of
$2.0\cdot 10^{-15}$ yr$^{-1}$.
