Install an undermount sink using silicone sealant and two-part epoxy to glue the sink to the solid countertop. Gluing the sink in place removes the need to drill into the bottom of the countertop, which weakens it.
Begin the process by cleaning the glue surfaces with a clean rag and acetone. Spread a 1/2-inch bead of the silicone sealant around the top of the sink, and lift it into place. After positioning it, use wooden braces and bar clamps to hold it while completing the rest of the installation process. Use a rubber mallet to tap the sink for any final positioning adjustments. Trim any excess silicone that squeezes from between the sink and countertop.
Cut small blocks from scrap countertop material. Mix the quickset epoxy according to the package directions, and apply it to the scrap blocks. Press the blocks into position so they span the edge of the sink. Allow the glue to dry for a minimum of 24 hours before proceeding with the installation.
Cut appropriately sized holes through the countertop for the faucet. Drop the supply lines through the holes, and screw the retaining nuts around the threads on the bottom of the fixture. Tighten them until snug, and attach the supply lines to the water valves. Install the drain lines to complete the process.