  We introduce new discretizations of the action for static quarks. They
achieve an exponential improvement (compared to the Eichten-Hill
regularization) on the signal to noise ratio in static-light correlation
functions. This is explicitly checked in a quenched simulation and it is
understood quantitatively in terms of the self energy of a static quark and the
lattice heavy quark potential at zero distance. We perform a set of scaling
tests in the Schroedinger functional and find scaling violations in the O(a)
improved theory to be rather small -- for one observable significantly smaller
than with the Eichten-Hill regularization. In addition we compute the
improvement coefficients of the static light axial current up to O(g_0^4)
corrections and the corresponding renormalization constants non-perturbatively.
The regularization dependent part of the renormalization of the b-quark mass in
static approximation is also determined.
