Rout very thin cracks using a hand-held electric router, because you need room to fill the marble crack with prepared epoxy. Clean all the areas of the crack thoroughly with a rag dampened with acetone.
Transfer the two components of the epoxy to a paper plate. Place them side-by-side so they do not touch.
Mix the two parts of the epoxy with a wooden popsicle-style craft stick. Add some marble dust the same color as the sill as you mix to tint the epoxy to match your marble. You usually can obtain marble dust from the same place you purchased your marble window sills or from most marble suppliers.
Fill the crack with the prepared epoxy using the craft stick. Work the epoxy deep into the crack with the tip of the stick. Keep adding epoxy until it slightly overflows the crack.
Smooth the epoxy surface with the flat part of the craft stick. Allow the epoxy to dry, undisturbed, for the amount of time specified by your particular brand of epoxy. This may vary from brand to brand.
Sand the epoxy with fine-grit sandpaper to remove the excess from the surface of the marble. Wipe the marble window sill clean with a rag. Sand the area again with extra-fine-grit sandpaper to restore the marble shine to the surface. Wipe away sanding dust with the rag.