  Defects which are predominant in a realistic model, usually spoil its
integrability or solvability. We on the other hand show the exact integrability
of a known sine-Gordon field model with a defect (DSG), at the classical as
well as at the quantum level based on the Yang-Baxter equation. We find the
associated classical and quantum R-matrices and the underlying q-algebraic
structures, analyzing the exact lattice regularized model. We derive
algorithmically all higher conserved quantities $C_n, n=1,2,...$ of this
integrable DSG model, focusing explicitly on the contribution of the defect
point to each $C_n$. The bridging condition across the defect, defined through
the B\"acklund transformation is found to induce creation or annihilation of a
soliton by the defect point or its preservation with a phase shift.
