This paper investigates a new low carbon iron-making supply chain planning problem in the steel industry under the carbon cap and trade mechanism, in which a steel company uses carbon emission quota to produce iron to meet determined demands over the planning horizon, and buys or sells the rights to emit carbon in the carbon trading market. The problem decides optimal carbon trade, raw material purchasing, raw materials and sinters inventory levels, as well as sintering and iron making production schemes, so as to minimize the total cost. A novel mixed integer programming model incorporating carbon emission reduction into operational decision making is developed and we propose a branch and price algorithm to solve it. The model is decomposed equally into a master problem and two subproblems. The branch and price algorithm is enhanced by two tailored techniques. First, the relaxation of the master problem is strengthened by introducing two families of valid inequalities. Second, we use a dynamic programming procedure to eliminate variables by path reduced cost. We test our algorithm on randomly generated data that simulate the practical production, and the computational results over the instances illustrate the effectiveness of the proposed algorithm. In addition, we derive the optimal operational decisions, and examine the impacts of carbon cap and carbon price analytically and numerically on total cost and carbon emissions. We make interesting observations based on numerical results and provide managerial insight to highlight the opportunity to reduce carbon emissions.