TCP is typically the default transport protocol of choice for its supposed reliability, even for message-oriented middleware (e.g., ZeroMQ) or inter-actor communication (e.g., distributed Erlang). However, under network issues, TCP connections can fail, which requires ensuring both at-least-once and at-most-one delivery at the upper middleware layer. Moreover, the use of TCP at scale, in highly concurrent systems, can lead to drastic performance loss due to the need for TCP connection multiplexing and the resulting head-of-line blocking. This paper introduces Exon, an oblivious exactly-once messaging protocol, and a corresponding lightweight library implementation. Exon uses a novel strategy of a per-message four-way protocol to ensure oblivious exactly-once messaging, with on-demand protocol-level "soft half-connections" that are established when needed and safely discarded. This achieves correctness, obliviousness, and performance, through merging and pipelining basic protocol messages. The empirical evaluation of Exon demonstrates significant improvements in throughput and latency under packet loss, while maintaining a negligible overhead over TCP in healthy networks.