On June 11, 1991 Alberro signed with the Texas Rangers as an amateur free agent. In 1991 he split time with the Rookie-league Gulf Coast League Rangers and the Advanced-A Charlotte Rangers. He went a combined 2–1 with a 2.75 ERA in 24 games, all in relief.\n', '

In 1992 he split time with Advanced-A Charlotte and the Class-A Gastonia Rangers. He went a combined 2–1 with a 1.91 ERA and 16 saves in 45 games out of the bullpen.\n', '

Alberro was promoted to the Double-A Tulsa Drillers in 1993 where in 17 games he compiled no decisions with a 0.95 ERA and five saves. His outstanding play soon saw him a promotion to the Triple-A Oklahoma City 89ers where in 12 games he did not involve in a decision, though his ERA did balloon to a combined 3.75.\n', '

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