C310 Pilot reported throttle control failed in flight. Pilot secured the affected engine and continued to a safe landing.

Narrative

I was approaching my destination and was given a descent to 5000' by ZZZ approach. To establish the descent; I reduced power to both engines; enrichened the mixtures; disengaged the autopilot altitude hold and pitched the nose down with the autopilot pitch wheel. Once established in the descent; I scanned the instruments and saw that the #1 engine was still at cruise power. I cycled the #1 throttle lever and there was no response from the engine; it remained at cruise power. Prior to this descent; both throttle levers were fully forward in cruise.I briefly considered 'throttling' the #1 engine with the mixture control. Attempting this created an unacceptable yaw as the #1 engine power delivery was either on; or off. I dismissed this idea. I decided to continue to my destination (about 30 miles) with the #1 engine at full power; then secure it prior to landing. I practice single engine landings during annual recurrent training. These landings are performed with the appropriate power and propeller settings per the POH to simulate single-engine flight. I have planned for a loss of engine power and made my standard procedure for a single engine landing is to fly an instrument approach with vertical guidance to the runway.ZZZ approach handed me off to ZZZ1 approach (ZZZ) as I was leveling off at 5000'. I reported in at 5000' and received the altimeter setting from ZZZ. After reading back the altimeter setting; I explained that I was unable to control the left engine and [advised ATC]. I requested radar vectors to a point 2-3 miles prior to the FAF of the RNAVXX approach to ZZZ1. My language on the radio was not this concise; but ATC understood. I secured the #1 engine as I was being vectored to the final approach course; about 2 miles prior to the final approach fix. The rest of the approach and landing were normal.The weather was clear; with winds gusting in the mid-20's; from 10-20 degrees to the right. In flight I remember it to be twilight but after landing I had to use a flashlight to gather my gear as I exited the airplane.After landing I removed the #1 engine cowling and saw that the throttle cable had separated (two pieces) where it was swagged to a metal rod that attached to the throttle crank at the engine. To the best of my knowledge; this cable is original to the airplane and is 50 years old. I have verified that replacement parts are not available from Cessna. My rationale for [advising ATC] was that the pattern at ZZZ1 was very busy. There are two flight schools on the field and the weather was excellent. My opinion is the greatest risk during this event would be to perform a single engine go around due to traffic.The good: 1. ATC service was excellent and the controller stated she would only give me turns into the operating engine. I did not expect ATC to have this knowledge. 2. My habit of practicing one instrument approach simulated single engine out when I practice approaches helped to make this situation more familiar. 3. My decision long ago to make use of an instrument approach with vertical guidance if I ever had to land with one engine inoperative. 4. I had recently briefed and practiced activation of a leg of the flight plan in my GPS. The activate leg function on Garmin GNS navigators is not intuitive.The bad:1. What I believe to be a 50 year old throttle cable failed at the engine end of the cable.2. The area where this cable failed is hidden and not able to be inspected visually.3. The cable is no longer available from the manufacturer.4. I may have no choice but to purchase a used cable from a salvage yard.