We analyze the amount of classical communication required to reproduce the statistics of local projective measurements on a general pair of entangled qubits, |Ψ>=sqrt(p) |00>+sqrt(1−p) |11> (with 1/2≤p≤1). We construct a classical protocol that perfectly simulates local projective measurements on all entangled qubit pairs by communicating one classical trit. Additionally, when 2p(1−p)2p−1log(p1−p)+2(1−p)≤1, approximately 0.835≤p≤1, we present a classical protocol that requires only a single bit of communication. The latter model even allows a perfect classical simulation with an average communication cost that approaches zero in the limit where the degree of entanglement approaches zero (p→1). This proves that the communication cost for simulating weakly entangled qubit pairs is strictly smaller than for the maximally entangled one.