Giffords is sending her resignation letter to House Speaker John Boehner, but the law dictates that representatives actually resign to their governors.
The U.S. Constitution does not allow appointments to the House. A lawmaker must be elected so if Giffords resigns this week, Brewer would presumably establish the dates of the primary and the special general election by next weekend.
According to state law, if the vacancy of a seat is more than six months from the general election, a special primary and general election must be held.
Once the governor declares the seat vacant, Brewer has three days to establish when she will hold the special election.
"Upon the congresswoman's office being officially declared vacant, I will call a special primary and general election in order to fill the 8th Congressional District vacancy, in accordance with Arizona law. The winner will complete the remainder of Congresswoman Giffords' term, which expires at the end of 2012," Brewer said in a statement Sunday.
"Congresswoman Gabrielle Giffords has been a noble public servant to the state of Arizona, and a model of what can be accomplished with persistence and determination. Her remarkable recovery over the last year is an inspiration to us all," she said.
Arizona law says the primary must be held 80-90 days after the declared vacancy and the general election 50-60 days after that.
That would put the primary around the end of April and the special general election around the end of June.
Currently, there are 242 Republicans in the House and 192 Democrats with one vacancy -- Oregon's 1st Congressional District, previously held by Democratic Rep. David Wu, who resigned in August. The special election to replace Wu comes in February.
With Giffords' resignation, there will briefly be two vacancies in the House, with 191 Democrats in office.
The following is text of Arizona statutes on special elections:
16-221 -- Special election to fill vacancy in CongressSpecial elections to fill a vacancy in the office of a representative in Congress shall be held only on the proclamation of the governor and for that purpose only.
16-222 -- Vacancy in the office of United States senator or representative
A. When a vacancy occurs in the office of United States senator or representative in Congress by reason of death or resignation, or from any other cause, the vacancy shall be filled at the next general election. At such election the person elected shall fill the unexpired term of the vacated office.
B. For a vacancy in the office of representative in Congress, if the next general election is not to be held within six months from the date of the occurrence of the vacancy, the governor shall call a special primary election and a special general election to fill the vacancy. The governor shall call the special primary election and establish its date within 72 hours after the office is officially declared vacant. Notwithstanding sections 16-313, 16-351 and 16-542, for a candidate for office at an election held pursuant to this subsection, the following apply:
1. The special primary election shall be held no less than 80 nor more than 90 days after the occurrence of the vacancy, and the special general election shall be held not less than 50 nor more than 60 days after the special primary election.
2. Nomination papers and nomination petitions shall be filed no later than 30 days after the date of the proclamation calling the election.
3. Any court action challenging the nomination of a candidate shall be filed no later than 5:00 p.m. on the fifth business day after the last day for filing nomination papers and petitions.
4. The superior court shall hear and render a decision within five days after the filing of the action.
5. Beginning 15 days before the date of the election, the county recorder or other officer in charge of elections shall mail early ballots within 48 hours after receipt of a complete and correct early ballot request from persons qualified to vote.
C. For a vacancy in the office of United States senator, the governor shall appoint a person to fill the vacancy. That appointee shall be of the same political party as the person vacating the office and shall serve until the person elected at the next general election is qualified and assumes office.
D. For a vacancy in the office of representative in Congress that occurs simultaneously with at least 100 additional vacancies in the office of representative in Congress as prescribed by 2 United States Code section 8, a special general election to fill the vacancy in this state shall be held no more than 49 days after the declaration of the vacancy unless a regularly scheduled general election or previously scheduled special general election is held within 75 days after the declaration of the vacancy.
16-223 -- Issuance of proclamation for special election governor; publication by boards of supervisors.
A. Within 10 days after a vacancy occurs in the office of representative in Congress, if a special primary and special general election are required by section 16-222, the governor shall issue a proclamation containing a statement of the time of the special primary election and the special general election and the offices to be filled.
B. The governor shall transmit a copy of the election proclamation to the clerk of each board of supervisors of the several counties.
C. The board of supervisors shall be notified by the clerk of receipt of the election proclamation, and within five days after its receipt the board shall meet and publish a copy of the election proclamation in an official newspaper of the county at least five days before the special primary election and at least five days before the special general election.
Fox News' Chad Pergram contributed to this report.