Many plastics, including ABS can be chemically welded. The result will be as strong as the original plastic. Thin parts may not have much surface area to join, and you may see a visible "melt" line at the repair.
Joining broken parts is essentially the same process as building things with a material like Lucite. You put the pieces together with a tight-fitting joint, then use a micro-pipette or syringe to apply solvent to the joint and nowhere else. Capillary action draws the solvent into the joint (which is why it needs to be tight-fitting, and why not much solvent is required). It isn't necessary to dissolve a huge amount of the material into a blob of plastic that runs together. The welding happens right at the mating surfaces.
For repairing broken parts that fit together tightly, a common technique is to fit the parts together, then use tape on one side to hold them in place. Apply the solvent on the other side. Once the joint is dry, remove the tape and check that side. On a really thick piece, the solvent may not wick all the way through the crack, in which case you can add a little solvent on that side.
If you're repairing broken parts that no longer mate tightly, you need a different method. You will need a filler material for the gaps. That's typically done by dissolving some of the same type of plastic in the solvent (most plastic cements are a mix of solvents and some dissolved resin; for welding, the cement needs to contain appropriate solvents and the same type of resin). Apply a thin layer to both broken edges, then push the pieces together.
This link, Intro To Solvent Welding Plastic, gives a list of suitable welding solvents for all the major types of plastics. I would test some of those first, though. For example, it lists acetone as a welding solvent for polyethylene and polypropylene, but my recollection is that dissolving would be too slow to be useful for welding. Similarly, methylene chloride is listed as not a solvent for ABS, but it's commonly the main ingredient of commercial ABS welding solvents. As for ABS, acetone, or a mix of acetone and other solvents, is commonly used for welding.
Many of the common solvents are available in the paint department in big hardware stores. There are also commercial solvent mixes sold in small bottles for this purpose. A little goes a long way, and the mix is optimized for specific types of plastics. But you'll spend almost as much for a few ounces of the proprietary magic mix as buying a big can of a specific solvent at a hardware store. That said, I've used this solvent and applicator from Micro-Mark pretty extensively, and both are great.