Date: 201510
Local time of day: 1201-1800
Location: ZZZ.ARTCC, US
Flight conditions: VMC
Light: Daylight
Aircraft: Citation X (C750)
Operator: Fractional
Operating under: Part 135
Flight plan: IFR
Flight phase: Descent
Airspace: Class A ZZZ
ATC / advisory: Center ZZZ
Reporter organization: Fractional
Reporter function: Captain; Pilot Not Flying
Reporter qualification: Flight Crew Air Transport Pilot (ATP)
Human factors: Communication Breakdown
Second reporter function: Pilot Flying; First Officer
Anomaly: ATC Issue All Types; Conflict Airborne Conflict; Deviation / Discrepancy - Procedural Clearance; Deviation / Discrepancy - Procedural Published Material / Policy
Detector: Person Flight Crew
When detected: In-flight
Result: Air Traffic Control Issued New Clearance; Air Traffic Control Issued Advisory / Alert; Flight Crew Took Evasive Action
Contributing factors: Procedure; Human Factors
Primary problem: Human Factors

Narrative: We were given a descent from FL360 to FL300; which we acknowledged; set; and confirmed. We were then given a new ATC frequency and I checked in with our altitude and 'descending to flight level 300' which was met with a 'roger' from ATC. We were descending through FL340 when ATC said 'traffic; 12 o'clock; 12 miles; opposite direction; flight level 330.' I responded; 'looking.' At the same time I was looking at the TCAS display and saw the aircraft 700 feet below (as we were still descending through FL337). Both the PF and I noticed the conflict and thought something didn't add up; we were descending head-on into the traffic. I quickly replied back and said 'verify [callsign] descending to 300?' ATC responded 'negative; you were given FL340; uh; expedite your descent through FL320'. The PF disconnected the autopilot and rapidly returned to FL340. Our lowest altitude while this was happening was FL334. While we were expediting our climb back to FL340; we heard one 'Traffic' alert from the TCAS. No TCAS RA was issued. We notified ATC that we had instead climbed back up to FL340 which contradicted his clearance in the heat of the moment and he said that was fine; thank you.I believe climbing was the better option; and I am happy that we as a crew thought the same thing. The PF responded quickly and assertively and did exactly what I expected of him; even though it was in conflict with the ATC instruction. I believe if we had indeed followed his instruction to expedite the descent; we would have had a closer conflict with the other traffic and likely a TCAS RA. Afterwards ATC queried what our instructions had been and I replied 'to the best of my recollection; we were cleared to descend to FL300.' He said 'yeah that should've been FL340' and said 'its ok now; no problem; thanks.'Once we landed; I called the phone number given to me by ATC. The controller who we were talking to during the event had contacted the previous controller we were working with before the hand off to request that we amend our descent instructions to FL340. The other controller accepted the amendment; but got busy with another aircraft on the frequency and never gave the amended instructions to us before he gave us the handoff. When we checked in with our cleared altitude of 'flight level 300' the controller missed that clearance.


Second narrative: [Report Narrative Contained No Additional Information]


Synopsis: CE750 flight crew experienced a traffic conflict during descent to an assigned altitude due to a controller error.