Why not? There's a Representative from Florida by the name of Alcee Hastings who was impeached, convicted and removed from office as a Judge. According to our current Constitution, he's not allowed to hold public office.
Here's how it's stated in Article I, Section 3, Clause 7: “Judgment in Cases of Impeachment shall not extend further than to removal from Office, and disqualification to hold and enjoy any Office of honor, Trust or Profit under the United States: but the Party convicted shall nevertheless be liable and subject to Indictment, Trial, Judgment and Punishment, according to Law.”
“According to our current Constitution, he’s not allowed to hold public office.”
That’s not true. In the case of Alcee Hastings, they removed him from office but did not disqualify him from any other office. The two don’t necessarily go together, it’s up to the Senate at the time.