We introduce and study the Doubly Balanced Connected graph Partitioning (DBCP) problem: Let G=(V, E) be a connected graph with a weight (supply/demand) function p:V →{−1, +1} satisfying|V1| }≤c s , for some constants c p and c s . When G is 2-connected, we show that a solution with c p =1 and c s =3 always exists and can be found in polynomial time. Moreover, when G is 3-connected, we show that there is always a 'perfect' solution (a partition with p(V 1 )=p(V 2 )=0 and |V 1 |=|V 2 |, if |V |≡0(mod 4)), and it can be found in polynomial time. Our techniques can be extended, with similar results, to the case in which the weights are arbitrary (not necessarily ±1), and to the case that p(V ) =0 and the excess supply/demand should be split evenly. They also apply to the problem of partitioning a graph with two types of nodes into two large connected subgraphs that preserve approximately the proportion of the two types.