  We present a scheme that allows integration of the Berry curvature and thus
determination of the Chern number of a qubit eigenstate manifold. Our proposal
continuously couples the qubit with a meter system while it explores a
quasi-adiabatic path in the manifold. The accumulated change of one of the
meter observables then provides an estimate of the Chern number. By varying the
initial state of the meter, we explore the delicate interplay between the
measurement precision and the disturbance of the qubit. A simple argument
yields a correction factor that allows estimation of the Chern number, even
when the qubit is significantly disturbed during the probing. The Chern number
arises from the geometric phase accumulated during the exploration, while we
observe the dynamic phase to produce a broadening of the meter wave function.
We show that a protocol, relying on three subsequent explorations, allows
cancellation of the dynamic phase while the geometric phase is retained.
