The Short Version
The claim is well-supported by authoritative sources. DARPA's own history, IEEE records, and multiple independent accounts confirm that ARPANET was developed under ARPA — a U.S. Department of Defense agency — with formal development, construction, and first operation occurring in the 1967–1969 timeframe. While earlier conceptual and planning work dates back to the early-to-mid 1960s, characterizing ARPANET development as "starting in the late 1960s" accurately reflects when the network itself was built and became operational.