Rep. Christopher Shays, a Connecticut Republican, also called for an investigation of his party's leadership.
The Republican National Committee took an active role in leading the charge against Shays-Meehan.
Ms. McMahon had 73% of the vote, to 27% for Mr. Shays, with almost all precincts reporting.
The Shays-Meehan bill would, for the most part, ban unregulated, soft money donations to the major parties.
He said the turmoil of Shays' Rebellion stirred up fears of mob rule among many leaders.
" But he predicted, "They'll have more success getting the discharge petition than getting votes for Shays-Meehan.
Representative CHRISTOPHER SHAYS (Republican, Connecticut): I happen to believe we're losing our moral authority to lead this place.
Those rules broke apart a package of adjustments to the Shays-Meehan bill opposed by the chamber's GOP leadership.
Like the McCain-Feingold bill in the Senate, Shays-Meehan would ban unregulated, "soft money" donations to the major parties.
But Ney defended his bill as a solid reform measure, and he criticized aspects of the Shays-Meehan legislation.
Mr. Shays had been a congressman for two decades before he was defeated in 2008 by Democrat Jim Himes.
Armey was referring to the campaign finance bill that passed the Senate in April, which mirrors the Shays-Meehan legislation.
As would the McCain-Feingold bill, Shays-Meehan would for the most part ban unregulated, soft-money donations to the major parties.
The House approved earlier versions of the Shays-Meehan bill in 1998 and 1999.
Shays and Meehan and their supporters in the Democratic leadership urged backers of the bill to vote against the rule.
"If they knew or should have known the extent of this problem, they should not serve in leadership, " Shays said Sunday.
DeLay is not damaged when a party maverick such as Christopher Shays of Connecticut says he thinks the majority leader should resign.
Shays-Meehan resembles the bill drafted by Republican maverick John McCain and Russ Feingold, a Wisconsin Democrat, that passed the Senate in April.
"This process is just truly an unbelievable outrage, " Rep. Christopher Shays complained.
Shays has wanted to see his bill become law for three years.
Instead, it turned into a fireworks display between Rep. Chris Shays, R-Conn.
"These e-mails are from some very interesting people, " Shays said, indicating that many of the messages could be appropriate to several committee investigations.
CNN: White House officials say e-mail problem was not a high priority at end of 1999
That is where Ney and Wynn feel they can knock down McCain-Feingold and Shays-Meehan, at least as far as undecided House members are concerned.
That's where Ney and Wynn feel they can knock down McCain and the Shays-Meehan bill, at least as far as undecided House members are concerned.
We've talked to -- been up and briefed and spent time with Congressman Chris Shays and the Republicans and Democrats on his committee as well.
An odd assortment of outside groups agree, including the National Rifle Association and the American Civil Liberties Union, which have aligned to combat the Shays-Meehan bill.
Heading into the primary, she led Mr. Shays by 68% to 20%, according to a poll released on July 31 by Public Policy Polling, a Democratic firm.
"What happened in Kenya strikes me as very likely to happen in Pakistan, and I don't know how we respond to it, " said Rep. Chris Shays, R-Connecticut.
The Shays-Meehan bill would ban soft money -- unregulated, unlimited funds to political parties -- and it would raise the limit on hard money contributions to individual candidates.
Rep. Chris Shays (R-Conn.), armed with audio tapes from the speaker's college course, told his colleagues the tapes were proof Gingrich had not been involved in partisan activities.
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