In this article, we have shown that a single noise acting on anyone of the qubits of an entangled two-qubit state can cause finitetime disentanglement. We have illustrated this fact by subjectingone of the qubits of a class of two-qubit X states and two-qubitpure states individually to three different kind of noises: amplitude, dephasing and depolarizing. While single-qubit channels ofamplitude and phase noises are shown to be ineffective in causingfinite time disentanglement in entangled pure states, single-qubitdepolarizing noise is shown to cause sudden death of entanglement even in pure states.