Automated hiring systems are among the fastest-developing of all high-stakes AI systems. Among these are algorithmic personality tests that use insights from psychometric testing, and promise to surface personality traits indicative of future success based on job seekers' resumes or social media profiles. We interrogate the reliability of such systems using stability of the outputs they produce, noting that reliability is a necessary, but not a sufficient, condition for validity. We develop a methodology for an external audit of stability of algorithmic personality tests, and instantiate this methodology in an audit of two systems, Humantic AI and Crystal. Rather than challenging or affirming the assumptions made in psychometric testing-that personality traits are meaningful and measurable constructs, and that they are indicative of future success on the job-we frame our methodology around testing the underlying assumptions made by the vendors of the algorithmic personality tests themselves. In our audit of Humantic AI and Crystal, we find that both systems show substantial instability on key facets of measurement, and so cannot be considered valid testing instruments. For example, Crystal frequently computes different personality scores if the same resume is given in PDF vs. in raw text, violating the assumption that the output of an algorithmic personality test is stable across job-irrelevant input variations. Among other notable findings is evidence of persistent-and often incorrect-data linkage by Humantic AI. An open-source implementation of our auditing methodology, and of the audits of Humantic AI and Crystal, is available at https://github.com/DataResponsibly/hiring-stability-Audit.