Antiretroviral therapy is not a cure for HIV-1 infection, as viral rebound inevitably occurs following discontinuation of treatment. The “block and lock” therapeutic strategy seeks to enforce proviral latency and durably suppress viremic reemergence in the absence of other medical intervention. The transcription associated cyclin dependent protein kinases (tCDKs) are required for viral expression from the 5' HIV-1 Long-Terminal Repeat (LTR), but the therapeutic potential of inhibiting these kinases for enforcing HIV-1 latency remains uncharacterized. Here we extended upon previous work to directly compare the effect of highly selective small molecule inhibitors of CDK7 (YKL-5-124), CDK9 (LDC000067), and CDK8/19 (Senexin A), and found each of these prevented HIV-1 provirus expression at concentrations that showed no toxicity. Inhibition of CDK7 caused cell cycle arrest, whereas CDK9 and CDK8/19 inhibitors did not, and could be continuously administered to establish proviral latency. Upon discontinuation of drug administration, HIV immediately rebounded in cells that had been treated with the CDK9 inhibitor, while proviral latency persisted for several days in cells that had been treated with the CDK8/19 inhibitor. Our results identify the mediator kinases CDK8/CDK19 as potential “block and lock” targets for therapeutic suppression of HIV-1 provirus expression.