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In 1963 Myers migrated to [[Rockwell International]], and the following year he began contract work for [[NASA]]'s space program. From 1964 he was the program manager of the [[Apollo program]]'s Command/Service Module Program, replacing John W. Paup.{{sfn|NASA, Myers|2004}} After a fire destroyed [[Apollo 1]] and killed three astronauts in January 1967, much of the program's management was purged; Myers, however, was retained.{{sfn|EA, Myers}} He migrated to the [[Space Shuttle]] program in 1969,{{sfn|NASA, Myers|2004}} soon after [[Apollo 11]]'s historic moon landing.{{sfn|EA, Myers}} Myers later described his work with Apollo as a highlight of his career.{{sfn|JSC, Dale Myers|1998}}
 
In 1970, Myers was promoted to Associate Administrator for Manned Space Flight at NASA, replacing [[George Mueller (NASA)|George Mueller]].{{sfn|Compton|1989|p=193}} In this position he stayed at mission control when [[Apollo 13]] experienced a crippling explosion;{{sfn|Compton|1989|p=388}} later he promoted the selection of geologist and astronaut [[Harrison Schmitt]] as geologista crew member forof [[Apollo 17]].{{sfn|Compton|1989|p=221}} He served until 1974.{{sfn|NASA, Myers|2004}} During this time he earned three [[NASA Distinguished Service Medal]]s, one in 1971 for his work on the Apollo program and two in 1974 (one for his work on [[Skylab]] and the Shuttle, the other for his work towards manned spaceflight).{{sfn|JSC, Dale Myers|1998}} In 1970 he also received an [[honorary doctorate]] from [[Whitworth University|Whitworth College]].{{sfn|JSC, Dale Myers|1998}} Myers was a president of the [[National Academy of Engineering]].{{sfn|National Academy of Engineering|2013}}
 
[[File:Apollo 16 meeting.jpg|left|thumb|300px|Myers, center-right, during a meeting to discuss whether [[Apollo 16]] should land on the moon (1972)]]