A burner includes a number of flame ports each having a fuel supply passage and provided on a pair of opposite walls of a combustion chamber such that each of the walls and the fuel supply passages define a cooling passage. In the burner, a wide stable flame region is achieved at a high excess air ratio, the amount of NOx produced is reduced and backfire is prevented by cooling the fuel supply passages. Furthermore, a flame area per unit area of each flame port is increased such that the burner can effect high-load combustion.