Short answer. Yes. The rule is explicit that a witness must answer to the fact of his or her previous final conviction for an offense. The general right to refuse answers that would degrade your reputation is stated first, and then this one category is expressly pulled back out of it.
What the law says
But a witness must answer to the fact of his or her previous final conviction for an offense.
Rule 132, Section 3 — Rights and obligations of a witness. Read the full provision →
An express exception to a general protection
The rule gives a witness the right not to give an answer that will tend to degrade his or her reputation. Admitting a past conviction plainly does that, so without more you might expect to be able to decline. The rule anticipates the point and closes it in the very next sentence, beginning with But. The structure is deliberate: the protection is stated, and then this one category is pulled back out of it. There is no room to argue that the admission is too damaging to be required.
What exactly must be admitted
The obligation is narrower than it may sound, and the limits are in the words themselves. It reaches the fact of a previous final conviction for an offense. It is the fact of the conviction that must be answered to. And it is a final conviction that the rule speaks of, which is not the same as a charge, an ongoing case, or an accusation that never resulted in one. Being asked about matters that never reached a final conviction is a different question from the one this sentence settles.
Why the rule draws the line here
A conviction is a matter of public record established by a court, not a private embarrassment. It is also a recognised way of testing whether a witness should be believed, which is the whole purpose of examining credibility. Allowing a witness to refuse would let the person best placed to know conceal a fact the court is entitled to weigh, while a witness with nothing to hide answers freely. Requiring the answer keeps the assessment of credibility from depending on whether the witness chooses to co-operate.
If you are due to testify
Knowing this in advance is better than meeting it cold in the witness box. Being required to admit the fact of a conviction is not the same as being required to relive the case, and the rule's other protections — against irrelevant, improper or insulting questions, and against examination on matters not pertinent to the issue — continue to apply to whatever follows the admission. If you expect to be asked, tell your lawyer beforehand so the position is prepared rather than improvised.