We consider a two-level quantum system (qubit) which is continuously measured by a detector. The conventional formalism, which implies the ensemble averaging, describes the gradual decoherence of the qubit state due to measurement. However, in each particular realization of the measurement process we can have the opposite effect: gradual purification of the qubit density matrix. This can be described by the recently developed Bayesian formalism suitable for individual quantum systems. The purification effect may be verified experimentally using present day technology and can be useful for quantum computing. In particular, the decoherence of a single qubit can be suppressed using continuous measurement and the feedback loop.