Motivated by an interesting algorithmic application, this paper is the first attempt to successfully design efficient approximation algorithms for the single-machine weighted flow-time minimization problem when jobs have different release dates and weights equal to their processing times under the assumption that one job is fixed (i.e., the machine is unavailable during a fixed interval corresponding to the fixed job). Our analysis shows that the trivial FIFO sequence can lead to an arbitrary large worst-case performance bound. Hence, we modify this sequence so that a new 2-approximation solution can be obtained for every instance and we prove the tightness of this bound. Then, we propose a fully polynomial-time approximation algorithm for the considered problem. The complexity of our algorithm is strongly polynomial.