Real-time multimedia applications such as VoIP and video conferencing require a minimum bit rate to ensure an acceptable level of quality of service (QoS) since they are delay-sensitive. In this regard, UDP is preferred to support the timeliness of real-time multimedia applications, but does not satisfy high video/audio quality when packet losses exist. Although TCP is generally used for elastic data traffic, it also can be used as an alternative to support better multimedia services due to its reliability. However, TCP is not suitable in that it chooses reliability over timeliness. This paper proposes a novel mechanism called TCP-MR (Minimum Rate) which simultaneously realizes timeliness and reliability by achieving end-to-end minimum rate guarantee over best-effort networks. To accomplish this goal, TCP-MR controls two congestion windows to reserve network bandwidth, adopts a persistent congestion detection algorithm to quickly react to continuous congestion state, and employs an enhanced loss recovery algorithm to shorten the recovery time of multiple packet losses. Only with modifications of the sender's side, we expect our proposal to be easily and widely deployed.