This paper reveals a stability-constrained approach for designing an odd-order recursive digital filter whose frequency bandwidths are freely tunable. The tunable filter has a cascade structure, and it also has definitely ensured stability. The odd-order tunable-bandwidth (OO-TBW) filter consists of a single first-order (1st-order) recursive section and a few second-order (2nd-order) recursive sections. Those sections are cascaded together to form the overall OO-TBW filter. To ensure that the cascade-form OO-TBW filter is definitely stable, a strategic technique is adopted, which expresses the coefficients of both the single 1st-order and multiple 2nd-order sections as a kind of specifically-defined composite functions. With such specific parameterizations, the resulting cascade-form OO-TBW filter is able to meet the stability requirement definitely. As a result, the resulting OO-TBW filter not only has an odd-order cascade structure, but its stability issue is resolved. To demonstrate the ensured stability and also the significantly high accuracy of the resultant OO-TBW filter, a lowpass OO-TBW filter (tunable-bandwidth lowpass filter) is designed, and the simulation results are detailed and analyzed.