We propose a new nonparametric test to determine whether finite-activity jumps are present in a discretely observed price process. For a univariate Itô semimartingale, we introduce the concept of censored increments for observations recursively sampled at exit times with a symmetric double barrier, and design a standardized test statistic to compare the sample moments of censored and uncensored increments. Simulation results show that our test has better finite-sample performance than other commonly used calendar time-based jump tests with a similar level of sampling sparseness, and is fairly robust to measurement errors including market microstructure noise and price staleness. Our empirical study provides strong evidence for the presence of jumps for 10 NYSE stocks in 2020, but the jumps are much less frequent than that suggested by some existing tests.