  We prove a version of Gauss-Bonnet theorem in sub-Riemannian Heisenberg space
$H^1$. The sub-Riemannian distance makes $H^1$ a metric space and consenquently
with a spherical Hausdorff measure. Using this measure, we define a Gaussian
curvature at points of a surface S where the sub-Riemannian distribution is
transverse to the tangent space of S. If all points of S have this property, we
prove a Gauss-Bonnet formula and for compact surfaces (which are topologically
a torus) we obtain $\int_S K = 0$.
