B737-800
Flight Crew Air Transport Pilot (ATP); Flight Crew Instrument; Flight Crew Multiengine
Flight Crew Last 90 Days 120; Flight Crew Total 14800; Flight Crew Type 11000
Aircraft Equipment Problem Critical; Conflict Airborne Conflict; Deviation - Altitude Excursion From Assigned Altitude; Deviation / Discrepancy - Procedural Clearance
Aircraft
Level cruise at FL390; brief engine rumble felt and then fine.  A few minutes later; sudden onset of vibration and rollback of N1 on engine #1.  I had to go to idle on engine #1 to reduced the vibration.  We notified ATC: 'ZZZ Center; we have had an engine failure; descending to FL220 and we will get back to you with the numbers'.  At our weight and with one engine at idle; I was unable to maintain altitude; so I had to start a slow descent to our max single-engine altitude which was FL220.  The Center Controller seemed a little bit slow to recognize our situation. I neglected to squawk XXXX; which may have helped the Controller recognize what we were saying.  We repeated the call in a second transmission and then a third after he advised us to maintain altitude with opposite direction traffic at 12 o'clock.  I could see the traffic visually and it was another transport aircraft offset slightly to the south of our track.  When the Controller did recognize our situation; he advised the approaching traffic below us to turn to the south; but the traffic was already behind us and no factor at the time. The engine continued to vibrate and following the guidance of the QRH; we shut down the engine and landed at the nearest suitable airport; which was ZZZ.  It was a clear day and a single engine visual approach was completed to a landing.  The ARFF (airport rescue firefighter) crews inspected the aircraft after we stopped on an adjacent runway and advised that there was oil draining onto the pavement.  After cleared to proceed and after shutdown; we confirmed that the engine had suffered some type of internal damage and was draining oil onto the ramp.
B737-800 Captain reported an engine failure during cruise.