Rudy Giuliani gave a big speech in Florida earlier today, and his remarks as prepared for delivery are here: joinrudy2008.com/article/pr/1085
[After my administration ends] We'll hand over a country where we've ended illegal immigration where it has to be done - at the border.
And also a country who's arms are wide open to people who come here, people who come here legally and openly. We want them, we need them, we will accept them, and what we want them to do, if they want to become citizens, is meet all the requirements and then be able to read English, write English, and speak English.
Ending illegal immigration won't be easy. It requires changing human behavior – but it needs to be done for everyone's good. I have the will. I have the way to do it, the plan to do it, and I have the track record to bring safety and order and fairness to a situation that is now [our] out of control. I've done it before, I can do it again.
I don't recall Rudy having stopped illegal immigration before, so I guess he's refering to his record as mayor of, what city was that again?, oh yeah, wasn't he the mayor of New York City, and on 9/11? I seem to recall him mentioning that.
The second paragraph can mostly be ignored. However, the claim that he can "end" illegal immigration at the border is fantastic for at least three reasons:
1. It's physically impossible to "end" illegal immigration. The most that could be done would be to sharply reduce it.
2. Since somewhere less than half of current illegal aliens are those who came here legally and then violated the terms of their visas, just concentrating on the border won't do anything about those who'd do the same in the future. Rudy might have a plan for that, but perhaps he should include that plan in his speech for completeness' sake.
3. The idea that illegal immigration can be reduced by concentrating only on border security is false, because unless we're going to construct a tall wall along all the borders, people will keep trying to come here as long as they know they can receive jobs and benefits. Rudy certainly has a plan for the former, but it involves the precursor to a national ID card. See also this.
As for his claims, his support for illegal immigration is well known; see for instance this video (youtube.com/watch?v=QhS-Ic9JohM) - from a September 7, 2001 radio speech - where he says that NYC is "quite tolerant of illegal immigration"... and he "agrees with that."
(On a sidenote, I typed in rudygiuliani.com only to find out that it's a parked page with ads; then I typed in "rudy" in Google and found out that his site is only in second place, behind the unrelated movie of the same name at imdb.com and just ahead of the Wikipedia entry about the same movie.)