The case-cohort design has been widely used as a means of cost reduction in collecting or measuring expensive covariates in large cohort studies. The existing literature on the case-cohort design is mainly focused on right-censored data. In practice, however, the failure time is often subject to interval-censoring: it is known to fall only within some random time interval. In this paper, we consider the case-cohort study design for interval-censored failure time and develop a sieve semiparametric likelihood method for analysing data from this design under the proportional hazards model. We construct the likelihood function using inverse probability weighting and build the sieves with Bernstein polynomials. The consistency and asymptotic normality of the resulting regression parameter estimator are established, and a weighted bootstrap procedure is considered for variance estimation. Simulations show that the proposed method works well in practical situations, and an application to real data is provided.