Stake string across the concrete surface. Use a spirit level to keep it perfectly horizontal. Position the string low enough so it just touches the highest point on the surface, allowing you to more easily see the pitch of the concrete.
Brace boards against the perimeter of the concrete surface if possible. Position the boards so the upper edges are level with the string. Keep the boards tight against the concrete with stakes driven just outside them and diagonal boards propping them up.
Thoroughly clean the concrete surface. Wash it with a grease-cutting detergent and scrape away any loose material.
Mix three parts sand and one part Portland cement to make a concrete suitable for pouring in thin layers. Mix cement and water separately, following the directions on the package. Spread a very thin layer of it on the existing concrete surface to help the old and new concrete bond.
Pour the new concrete onto the old surface until it is level with the board edging. Wait 48 hours before removing the board supports.