The real trick here is to ensure that the new wood you select to border the concrete is suited for this application - specifically earth-to-wood contact. We'll bet that the material originally installed was red-wood. Redwood is not only attractive as a border to concrete, but it has certain innate properties that permit it to resist deterioration from fungus or pest infestation. Still, with prolonged exposure to the elements, even redwood deteriorates.
Although redwood has served well in a variety of applications, improved technology has given us materials that are pressure-treated with chemicals specifically designed to withstand contact with earth or concrete.
Start your repair project with a trip to the lumber yard for a pile of pressure-treated fir. While there, ask for a tube of epoxy or some concrete glue. You'll want to apply a hardy portion of the adhesive to both the face of the board and the concrete. It's vital that before you start gluing you cut the new trim board to size and test the fit. In order to achieve a good permanent bond, make sure that both the concrete and the wood are completely dry and free of any surface debris.