Unit root test statistics may not have the usual asymptotic properties when the variance of innovations is unstable. In particular, persistent changes in volatility can cause the size to differ from the nominal level. We propose a class of modified unit root test statistics that are robust to the presence of unstable volatility. The modification is achieved by purging heteroskedasticity from the data using a kernel estimate of volatility before the application of standard tests. In the absence of deterministic trend components, this approach delivers test statistics that achieve standard asymptotics under the null hypothesis of a unit root. When the data are homoskedastic, the local power of unit root tests is unchanged by our modification. We use Monte Carlo simulations to compare the finite sample performance of our modified tests with that of existing methods of correcting for unstable volatility.