Causal indirect and direct effects provide an interpretable method for decomposing the total effect of an exposure on an outcome into the indirect effect through a mediator and the direct effect through all other pathways. A natural choice for a mediator in a randomized clinical trial is the treatment's targeted biomarker. However, when the mediator is a biomarker, values can be subject to an assay lower limit. The mediator is affected by the treatment and is a putative cause of the outcome, so the assay lower limit presents a compounded problem in mediation analysis. We propose two approaches to estimate indirect and direct effects with a mediator subject to an assay limit: (1) extrapolation and (2) numerical optimization and integration of the observed likelihood. Since these estimation methods solely rely on the so-called Mediation Formula, they apply to most approaches to causal mediation analysis: natural, separable, and organic indirect, and direct effects. A simulation study compares the two estimation approaches to imputing with half the assay limit. Using HIV interruption study data from the AIDS Clinical Trials Group described in Li et al 2016, AIDS; Lok and Bosch 2021, Epidemiology, we illustrate our methods by estimating the organic/pure indirect effect of a hypothetical HIV curative treatment on viral suppression mediated by two HIV persistence measures: cell-associated HIV-RNA and single-copy plasma HIV-RNA.