The 1961 NFL draft took place at the Warwick Hotel in Philadelphia on December 27–28, 1960. The league would later hold an expansion draft for the Minnesota Vikings expansion franchise. This draft was also the first regular draft for the Dallas Cowboys as they had only participated in the 1960 NFL expansion draft that year.
The Cowboys held the worst record in the NFL the previous season, but selected second in this draft because the expansion Vikings were awarded the first overall pick. The Vikings used that pick to select running back Tommy Mason.
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Text from the Wikipedia article “1961 NFL Draft”, available under CC BY-SA 4.0.