Kill the established lawn first with a rototiller. You do not need to use an herbicide since St. Augustine sod will suffocate any lawn trying to come back.
Rake out the soil with a garden rake all the dead grass and debris.
Test your soil for nutrients for nitrogen, phosphorus, potassium and pH.
Add the soil amendment with a wheelbarrow according to your test results. Till the amendment in with a rototiller.
Grade the soil with a landscape rake till the appropriate grade is reached.
Water roll the soil till firm with a water roller, which you can rent from the hardware store for about $10 per day. If one is not available you can water the soil down to the depth of 2 to 3 inches; then let it settle for a week.
Lay the St. Augustine sod strip. The first course will all be whole pieces.
Cut in half the first piece of the second course or row with a sod knife. Sod should be laid staggered for strength. The third row will start with a whole piece. Continue in this manner until you're done.