  We consider a magnetic Schr\"odinger operator $(\nabla^X)^*\nabla^X+q$ on a
compact Riemann surface with boundary and prove a $\log\log$-type stability
estimate in terms of Cauchy data for the electric potential and magnetic field
under the assumption that they satisfy appropriate a priori bounds. We also
give a similar stability result for the holonomy of the connection 1-form $X$.
