I have failed to find any evidence whatsoever that the phrase was said by Barnum's rival in the Cardiff Giant hoax.
Chicago has claimed that the phrase is from Michael "King Mike" Cassius McDonald, but the digitized Chicago Tribune doesn't turn up a single citation at any date.
What we know for certain is that the phrase was popular with New York City gamblers in 1882-1883. Variants (using "flat" and "fool" for "sucker") of the phrase, however, date to much earlier.
Wikipedia: P. T. Barnum
Phineas Taylor Barnum (July 5, 1810 — April 7, 1891), American showman who is best remembered for his entertaining hoaxes and for founding the circus that eventually became Ringling Brothers and Barnum and Bailey Circus.
About ten days after the discovery, and about the time the Cardiff Giant, as the papers had named it, started receiving national attention, Hull sold two-thirds interest in the giant for $30,000 to a five-man syndicate in Syracuse, the head of which was a banker named David Hannum. The syndicate moved the giant to an exhibition hall in Syracuse and raised the admission price to a dollar a head. Unknown to them, P. T. Barnum sent an agent to see the giant and make an assessment. The particular Sunday the representative saw the giant, the crowds were abnormally large -- about 3,000 people. The agent wired the news back to Barnum and Barnum instructed him to make an offer of $50,000 to buy it. Hannum turned his offer down.
The Cardiff Giant was the most talked about exhibit in the nation. Barnum wanted the giant to display himself while the attraction was still a hot topic of the day. Rather than upping his offer, Barnum hired a crew of workers to carve a giant of his own. Within a short time, Barnum unveiled HIS giant and proclaimed that Hannum had sold Barnum the original giant and that Hannum was now displaying a fake! Thousands of people flocked to see Barnum's giant. Many newspapers carried the version that Barnum had given them; that is, Hannum's giant was a fake and Barnum's was authentic. It is at this point that Hannum -- NOT BARNUM -- was quoted as saying "There's a sucker born every minute." Hannum, still under the impression that HIS giant was authentic, was referring to the thousands of "fools" that paid money to see Barnum's fake and not his authentic one.
Made in Chicago: "There's a sucker born every minute."
Irish immigrant Michael ("King Mike") Cassius McDonald, who ran Chicago's first crime syndicate, is said to have coined this phrase. According to legend, it was his explanation for how he planned to get enough customers for his gigantic, four-story gambling house. This casino-like palace, "the Store," was located close to City Hall, and it provided a gathering place for Democratic politicians as well as unsuspecting gamblers. McDonald obtained the cooperation of the police force, politicians, and an army of skilled confidence men to run his rigged games.
McDonald's criminal activities prefigured those of Al Capone and other Chicago gangsters.
King Mike is also credited with saying, "Never give a sucker an even break."
16 July 1846, Boston (MA) Daily Atlas, "Etchings in England: An English Sporting Scene -- The Derby Day," col. B:
And so shall the game go on from generation to generation; for, spite of the police, and the warnings of wisdom, there is truth, undying truth, in the proverb, that "there is a flat born every minute."
19 August 1857, New York (NY) Herald, pg. 4:
The confidence of these gentlemen in the gullibility of human nature must be very strong, and they have undoubtedly adopted the maxim that there is a new fool born every day.
8 August 1875, Indianapolis (IN) Journal, "Wall Street Wiles," pg. 3:
Now I would suggest that some of these parties should do the state some service upon the pleasant islands adjoining Manhattan, and thus be prevented from following out their business, which is nothing more or less than deluding the unwary and trading upon the credulity of a mass of gullible fools who never die and one of whom is born every minute.
7 January 1882, Daily Inter Ocean (Chicago, IL), "Keep Your Money -- People Would Do Well to Look Carefully before They Leap," pg. 7, col. 3:
It was an expert confidence man who said that "there was a sucker born every minute, and some have remained suckers all their lives."
5 February 1883, Cincinnati Enquirer, pg. 4, col. 4:
"A Sucker Born Every Day"
The Game in This and Other Cities - Harry Withrow's Arrest in Boston.
A reporter asked a well-know bunko man one day how it was that people still continued to be swindled by the same old process when the papers are constantly warning them. The reply was that there was a sucker born every day. And so it seems, for the old time scheme continues to be worked with success all over the country.
28 March 1883, Atkinson (KS) Globe, "The Gamblers of New York," pg. 3, col. 1:
"Gamblers," he (J. H. Stanley - ed.) continued, "call all these outside the fraternity ''suckers.' They claim that a 'sucker' is born every minute, and that New York is the greatest place in the world for them."
30 December 1883, New York Timespg. 4:
"'There's a sucker born every minute,' as the gamblers say, and it is only by these weaklings that our disreputable friends here live."
Plain Truths About Stock Speculation:
How to Avoid Losses in Wall Street,
With a Visitors' Directory In and Around New York
Brooklyn, N. Y.
(No author, but "Copyright by E. V. Smith, Proprietor" - ed.)
The saying, "A fool is born every minute," is said to have originated with one of the most prominent operators in Wall street. Certain it is that a new crop of fools is always counted upon and seldom fails to respond to the allurements offered.
8 April 1890, Sandusky (OH) Daily Register, pg. 2, col. 3:
P. T. Barnum once remarked that the American people like to be humbugged, and the success that crowns the efforts of spiritualistic mediums, faith cure healers and others of that ilk in this country demonstrates the correctness of Barnum's statement. It is an old and homely saying that "there is a sucker born every minute," and if that be true there is no danger of the supply being exhausted.
28 September 1906, New York Times, pg. 8:
From The Evening Mail.
He will get all the votes coming to a political Barnum whose motto is that of the original Barnum: "A sucker is born every minute."
12 November 1932, Hollywood (CA) Herald, pg. 13 ad:
HE COULD SELL SNOWPLOWS...IN HAWAII!
There may be a sucker born every minute...but the day Jimmy Bates was born...the clocks weren't running.