Not sure how I stumbled into this site but as long as I'm here......
In short yes, there is hope. However, without knowing the context of your situation I can offer perhaps some suggestions -mostly echoing a lot of the solid advice from others.
I have a PhD and have served on both pre-doctoral internship and post doctoral residency admission committees. While the selection criteria and processes that I am familiar with may differ substantially from those used in the academic selection processes for entrance into a doctoral program (which may be more heavily weighted toward quantitative variables and initial algorithmic cuts (rather than research and clinical work), I think several suggestions offered by previous posters are important.
1) In your CV acknowledge that you withdrew (not dismissed) from your previous academic institution 2/2 medical issues. I'd be vague and omit that you were asked to withdraw; If you get to an interview stage (assuming there is one) then a more detailed disclosure will be likely required. You need to get past the first hoop.
2) You indicate a high level of competence in the program prior to the honor violation, emphasize those accomplishments on your CV
3) Speak with your major/dissertation chair or advisor; unless things have changed radically over the past 30 years, they still ask for letters of recommendation and these will be important. Let them tell your story and also highlight your strengths and accomplishments. Of couse you could always go back to the undergraduate well and ask your profs from that institution write letters again.
4) Be prepared to explain how the disorder contributed to this anomaly in your behavior; what was the functional impairment of the disorder/illness (i.e., judgement, attention). How has that functional impairment be mitigated, resolved managed etc). So if you had an episodic mood disorder that impaired your judgement, how do you manage sx now? Even if the plagiarism was really egregious but clearly linked to a significant alteration in thinking and bx, you may get a pass.
On a side note, no forensically oriented mental health expert would ever vouch "that it will never happen again." (unless you're dead or have some some other countervailing factor :-) ). No such thing as a guarantee. I'll leave you with this gem from Tommy Boy
Tommy: Here's the way I see it, Ted. Guy puts a fancy guarantee on a box 'cause he wants you to feel all warm and toasty inside.
Ted Nelson: Yeah, makes a man feel good.
Tommy: 'Course it does. Why shouldn't it? Ya figure you put that little box under your pillow at night, the Guarantee Fairy might come by and leave a quarter, am I right, Ted?
Ted Nelson: What's your point?
Tommy: The point is, how do you know the fairy isn't a crazy glue sniffer? "Buildin' model airplanes!" says the little fairy, well, we're not buying it. He sneaks into your house once, that's all it takes. The next thing you know, there's money missing off your dresser and your daughter's knocked up, I've seen it a hundred times.
Ted Nelson: But why do they put a guarantee on the box?
Tommy: Because they know all they sold ya was a guaranteed piece of shit. That's all it is, isn't it? Hey, if you want me to take a dump in a box and mark it guaranteed, I will. I've got spare time. But for now, for your customer's sake, for your daughter's sake, ya might wanna think about buying a quality product from me.