Joint decoding is employed in many coding schemes to achieve a larger rate region. However, decoding complexity escalates considerably when joint decoding is used. This paper studies the achievable sum-rate of the two-user Gaussian interference channel when joint decoding is replaced by successive decoding. When interference is weak, it is shown that, for a wide range of transmitters' powers, rate-splitting and joint decoding are not required. A single split scheme with successive decoding achieves the Han-Kobayashi sum-rate. When rate-splitting is required, a novel rate-splitting scheme is proposed that does not use joint decoding. The number of required splits and the amount of power allocated to each split are expressed in closed forms. It is proved that the difference between the sum-rate of this scheme and that of the Han-Kobayashi scheme is bounded, even when transmitters' powers approach infinity.