The Dallas Cowboys did not participate in the NFL Draft in December 1959. The team instead chose players in an expansion draft that took place later.
The first NFL draft in which the Cowboys participated was the one during the offseason between the 1960 and 1961 seasons. Unlike modern drafts, those old drafts took place earlier in the year.
The 1961 Draft actually took place on December 27 and 28 in 1960. The draft then was not the spectacle it is today, but it did receive news coverage.
Dallas had traded its rights to the second overall pick to Washington in exchange for the rights to Eddie LeBaron.