Purchase a treated-wood planter box that's equal to the desired inside dimensions of the wishing well fountain. A galvanized plastic tub also works for small wishing well fountains.
Excavate and level an outdoor area that fits the footprint of the wood planter box, plus a 6-inch extension on each side for the rocks. Center the planter box on top of the excavated area.
Cut a section of heavy plastic pond liner to fit inside the box, from rim to rim, all the way around. Staple the plastic in place around the top of the planter's rim.
Put on a dust mask and follow the package directions to mix a bucket of thinset mortar. Spread a 1/2-inch-thick layer of thinset along the excavated ground on one side of the planter box. Press medium-sized river rocks or other rocks into the mortar. Follow the process to add a row of medium-sized rocks around the base of the planter.
Spread a 1/2-inch-thick layer of thinset on top of the rocks on one side of the box. Stack and press medium-sized rocks into the mortar; center the second row of rocks between two rocks on the first row. Staggering the rocks makes a stronger wall.
Continue the process to add a second layer of stones to each side of the box. Follow the process to add as many layers of rock as necessary to cover the sides of the box. Always stagger the rock placement on subsequent rows. Fill in with small river rocks.
Place a concrete block in the center of the inside of the box. Attach a fountainhead to the spout on the fountain and extend it all the way. Sit the pump on top of the concrete block. The fountainhead should extend to within 1 inch of the box rim. If not, stack another block on top of the first to elevate the platform.
Drape the power cord from the pump over the back side of the box. Spread thinset along the rim of the box; do not cover the cord. Press small river rocks into the thinset along the rim. Follow the process to cover the remaining three rims with rocks. Allow the mortar to dry overnight.
Fill the planter box with water. Run the power cord to an approved outdoor outlet and plug it in to start the pump. Adjust the fountainhead so that water stays inside the wishing well.