Start by finding the rock you want to use for your basin/birdbath. make sure it does not have obvious cracks as that will be where your project breaks later in construction.
Step 1: Marking Your Bowl
Use chalk or a scribe to mark the outside edge of the bowl you will be creating
Step 2: Right Tools for the Job.
Start this project bu using an angle grinder (in this case 115mm grinder) fitted with a wet and dry stone cutting diamond edge steel blade to cut the stone along the chalk or scribe line you drew on the rock to permanently mark it. In addition to the grinder you will need a strong thin stone chisel and reliable hammer. Please make sure you have a good pair of safety glasses as the stone shards (granite in my case) are as sharp as glass and can easily blind you if it ends up in your eyes.
Step 3: Cutting and Removing Rock to Make the Bowl.
Once you have cut the outline of your bowl, start cutting lines as deep as the blade will go in the rock within the outline you created. Shallower at the edges and deeper towards the middle. Using your hammer and chisel knock these slats of stone out (wear your safety glasses here).
Once an entire layer has been removed, repeat the process but cut diagonally to the previous cuts and chisel them out.
It's dusty work, as you can see. a Dust mask or hasmat mask is a good idea.
Step 4: Smoothing and Finishing the Bowl.
Once you have removed the bulk material from the rock and reached the depth you desire its time to smooth the bowl.
using the same diamond edge steel blade start smoothing the rough break and chisel marks left by the tools.
It remains a dusty job,and should only be attempted outside.
If you happen to own a wet stone grinder attachment this step will go quickly. (I don't so i did it with the diamond blade)
keep going until you are satisfied with the smoothness.
Step 5: Finished Product.
This item is a birdbath but if a basin is required a 40-50mm stone hole saw can be used to drill a drain hole for the bowl.
Happy cutting and creating.