The fact is that amnesty for illegal aliens already exists under certain conditions. Here is what the current law says:

http://en.wikipedia.org/wiki/Cancellation_of_removal

Cancellation of Removal for Non-Permanent Residents

Pursuant to INA §240A(b), cancellation is available to a non-permanent resident of the United States in any immigration status who--

1) Has continuously resided in the United States for at least ten years; and
2) Has been a person of good moral character throughout this time; and
3) Is not otherwise subject to criminal bars arising from a conviction of any crime outlined in INA §212(a)(2), §237(a)(2), or §237(a)(3); and
4) Establishes that removal would result in "exceptional and extremely unusual hardship" to the alien's spouse, parent, or child who is a United States citizen or legal permanent resident.

The application form for non-LPR-based Cancellation of Removal is the EOIR-42B.