In this paper, we study a school choice problem in which students' preference lists and schools' priority orderings are strict. We introduce a new mechanism, the priority secure Boston mechanism (PsBM), under which each school secures its quota of students with the highest priorities--ensuring that they are accepted whenever they apply--following the requirement introduced by the secure Boston mechanism (sBM) (Dur et al., 2019). Under PsBM, unsecured priorities depend on the timing of applications, whereas under sBM they depend on the rankings of schools in students' preference lists. Our theoretical results show that PsBM is less manipulable and more stable than the Boston mechanism (BM), and is not Pareto dominated by BM. Furthermore, we compare PsBM with sBM and show that, in theory, the two mechanisms are not comparable in terms of manipulation, stability, and Pareto efficiency. Numerical simulation results show that, on average, there are fewer manipulating students, fewer students involved in blocking pairs, but also fewer students matched to their first choice schools under PsBM than under sBM. Finally, we extend PsBM to accommodate general secure structures, under which each school can secure an arbitrary set of students, and we briefly discuss the role of hierarchical secure structures, under which a student is secured by a school only if every student has a higher priority is also secured, in preventing manipulation.