Our shower has a has sill about 4" above floor level made from a single piece of marble, approximately 60" x 6" x 1". The shower surround runs down the center, and is well-sealed to it, with no evidence of seepage. However, there is a wide run of soft limestone on the outside which is eroding away, leaving a lengthwise crack in the marble; this has been growing increasingly wide and deep over the past few years.
I'd like to clean it out and fill it with something epoxy-like (ideally black to match the dominant color of the marble), but I'm wary that anything that expands during curing might cause further problems with the stone. Are there any materials that are well-suited to this?