No. If a U.S. Citizen marries someone who is undocumented, the best the alien can hope for is Permament Resident Status (a green card) and citizenship three years after the date of adjustment. If the person is "illegal" because (s)he entered with inspection and overstayed and/or worked without authorization, (s)he may be entitled to adjust his or her status and have his or her unauthorized employment and unauthorized presence forgiven. If however, someone is "illegal" because he or she entered the U.S. without inspection (such as by sneaking across the Mexican border), such relief would most likely not be available unless the alien would be eligible under section 245(i).
Having said this, the marriage would have to be bona fide, meaning a true marriage, not one entered into for the purpose of evading U.S. immigration laws. It is best to consult with a qualified immigration lawyer to discuss the specifics of your case and to determine eligibility. Good luck!
Although I agree with the answer provided by my colleague, there are citizenship venues available to a person who entered the US without inspection ("EWI"), i.e. illegal undocumented alien. Such types of waivers are very case specific and you should consult an immigration attorney to discuss your circumstances. Once a waiver is granted, an alien may reenter the US legally and eventually file for the US citizenship. Generally, an undocumented illegal alien may file for a waiver based on a hardship to the US citizen spouse. This waiver can only be filed from the outside of the United States. An illegal alien will have to leave the US and go back to Mexico. Since this waiver is discretionary it may get denied if US immigration services finds that the hardship to the US citizen spouse is not extreme enough to admit the alien back into the US. In addition, the correct answer also depends on the amount of time an alien spent in the US undocumented. For example, if an alien has been previously removed and reentered the US without inspection again, or even if he reentered the US without inspection multiple times. In such cases, a waiver may not be available until after an alien spends 10 years outside of the US.
There is also an option of cancellation of removal if the illegal alien gets detained by the USCIS. If an alien has been in the United States for at least ten years and if his removal/deportation would cause exceptional and extremely unusual hardship to US citizen spouse, parent or child who is a US citizen or permanent resident, he may qualify for Cancellation of Removal. It is very difficult to prove such level of hardship, but if an alien is granted Cancellation of Removal, he will be eligible for permanent residence. Please contact an experience immigration attorney to discuss possible resolutions of your particular case. The responses and information are intended to be general and should not be relied upon for any specific situation. Best of Luck!