It would require 1) a constitutional amendment (yeah, I know.), 2) amnesty, permanent residency, or some similar status granted to all current undocumented residents with no criminal record. 3) and an NGO to register all current undocumented residents.
The amendment and amnesty would be a quid pro quo to get broad support. Specifically, the 14th amendment would be further amended to no longer provide citizenship to "all persons born or naturalized in the United States," starting the moment the new amendment is passed, in exchange for an amnesty (or similar status) for all current residents violating current immigration law.
Since the current law gives citizenship to anyone born on US soil, eliminating that should give the "conservatives" something they like while the "liberals" should like the amnesty or residency status.
Considering the time frame required of constitutional amendments, an NGO would be charged with registering and creating a database of all current "undocumented migrants" up to a certain date. After the passage of the amendment, those names would be submitted to U.S. government for amnesty.