The Kronig-Penney problem is a textbook example for discussing band dispersions and band gap formation in periodic layered media. For example, in photonic crystals, the behaviour of bands next to the band edges is important for further discussions of such effects as inhibited light emission, slow light and negative index of refraction. However, the standard Kronig-Penney equation does not explicitly state the band edge conditions. This paper derives a new solution for the Kronig-Penney problem that explicitly displays the band edge conditions as well as contains all other essential physics of band formation. Therefore, the present exposition should show the student that the band edge conditions are not simply special cases of the familiar Kronig-Penney equation but, instead, are an integral part of the band theory. For the computationally minded student, the new equation is particularly convenient for calculating the positions of closely spaced band edges. The present results can be taught alongside the Kronig-Penney equation in advanced undergraduate or beginning graduate quantum mechanics, solid state theory and photonics courses dealing with wave propagation through periodic layered media. (Contains 4 figures and 1 table.)