Marble is very heavy and is rock, but also is very soft and can be worked with wood working tools. A broken marble slab is pretty expensive to have repaired too.
I should mention very first that the highly polished reflective shine on some marble can't be duplicated at home. It takes some pretty expensive equipment and diamond dust, among other things, to achieve the high shine. A repair to marble of this type may not be perfect, but if you've priced the repair and it's too expensive the repair would probably look better than the piece being broken off.
The repair may be noticeable on a less shiny marble too, but is easier to conceal than on the shiny stuff.
You'll need a perfectly flat work surface a little larger than the marble slab. A piece of 3/4 inch plywood on saw horses works well.
Clamps are an absolute must. Pipe clamps are reasonably priced and are sturdy and easy to work with. You buy the clamp ends, then buy threaded pipe the length you need from a plumbing shop. You'll need two clamps, one to clamp the full length of the slab and one to clamp the width.
Marble will generally break along the line of the design in the marble, so the break will be irregular and will be along a variation of colors. The irregular break line will make it easier to get the piece to go back where it's supposed to be and the color variations will make it easier to conceal the repair.
If you have a stone works or cemetery marker business nearby you may be able to buy colored epoxy glue from them. If you do have such a business available, take the smallest piece of marble, if possible, in for a color match.
If you have to buy from a hardware store see if they may have different colors of epoxy glue, if not try for polyurethane glue, you may be able to find it in colors. If you can't find a near color, buy colorless clear epoxy or polyurethane glue.
The glue is gap filling, so you'll need to use care with the amount of glue you use. You need enough to do the glue job, but not so much that it will add so much area that there will be a noticeable jog along the sides of the slab where the two pieces meet.
Make a practice run first and dry fit the broken pieces. Clamp them up so you can tell just where the clamps need to go to be the most effective. Check to be sure that the top is exactly even and flat.
Once the glue and clamps are applied you've got a lifetime repair, so it needs to be as exact as possible, because you won't be able to sand it down to make everything smooth.
Apply glue to both surfaces with a brush so you can be sure that it gets into all the nooks and crannies. Don't apply glue to the top 1/8 inch, you'll need room to do your final blending and repair hiding. The glue may squish up when you tighten the clamps, so keep any excess squished out cleaned up and uses a toothpick to scrape enough glue out of the crack to give you room for the final touches. This is easiest done if you have a helper, so one can do the clamping and squishing while the other does the squish wiping and cleaning.
If the glue is the right color don't scrape any out of the crack, just level it even with the rest of the top as much as you can. Leave a little dip rather than a hump. You can add to the dip if necessary, but it could be very difficult to flatten the hump.
The rest is slow patient work. Leave the repair clamped up for at least 12 hours to let the glue cure well. If you have a clear glue use an artists brush and acrylic paint which is a color as near to the marble as possible and paint inside the repair crack. Let the paint dry completely, then work more of the clear glue into the crack, keeping away from building a hump.
Let everything dry completely then use a paste wax, such as Antiquax rubbing back and forth across the repair with many light applications of wax to fill in any voids, then wax the whole top equally.
You may be able to see where your repair is, but don't tell anybody else and I bet they won't notice it. It won't be quite as perfect as a professional with expensive tools would do, but it's good job and you can be proud of doing it yourself.