The man, Nick K. Gates, was charged with threatening to burn Cruz and Cruz's father if the senator from Texas did not pay Gates $3 million, The Houston Chronicle reported.
Gates was being held in Harris County Jail with a bail of $10,000 as of Tuesday.
The arrest warrant for Gates says that Cruz's San Antonio office received a phone message where the caller said that he "would kidnap, murder, and burn Ted Cruz and his father."
"Ted has a choice. Give me three million dollars or lose the sun," the voice in the message said, according to the arrest warrant. The caller added that "due to government misconduct the sun would blow up" and said "he might be able to prevent the sun from blowing up if he receives three million dollars."
Gates allegedly left the message from his home telephone. On June 10, Gates admitted to making the phone calls, according to prosecutors.
According to the arrest warrant, Gates also was previously convicted of a felony for attempted retaliation for threatening a Houston police officer and the officer's child after the office arrested Gates for driving while intoxicated.