If you have composed a setting and plan to use it in any Roman Catholic liturgy you must first get the approval of your local conference of Catholic bishops.  In the United States that conference is the USCCB.  The conference is a large organization, and the part that handles liturgical music text approval is the Committee on Divine Worship.  My contact there was gracious, accommodating and proficient, especially given that the committee must have been deluged with requests around the release of the new Roman Missal.  To request approval, send an email to the Committee on Divine Worship in your local conference (the US address is bcl@usccb.org.)  Include your name and and a short cover on what you are requesting to be approved,  something like "using excerpts from English translation of the 2010 ICEL Roman Missal".  If your setting includes Psalm responses or Gospel verses then include "using excerpts from the English translation of the 1969, 1981, 1997 ICEL Lectionary for Mass".  Attach a PDF of the score, including lyrics.  Be sure to replicate the exact word sequence, spelling, capitalization and punctuation - I used this online reference.  You are allowed to repeat an immediately preceding phrase, but don't go overboard.  Place the copyright and approval notices in the score as if it were already approved so they can be edited too.  The process will take a few weeks, but no matter how many times you may have checked, there will be revisions.  Here is the approval for Missa ex Corde:

ICEL permission

After getting approval from your conference of bishops, you will also need permission to use the texts from ICEL which holds the copyright on the 2010 Roman Missal and the 1969, 1981, 1997 Lectionary for Mass.  To request permission, send an email similar to the conference of bishops email with a PDF attachment of the score, including lyrics, to permission@eliturgy.org.  Mention that you already have approval from your conference of bishops.   Again, my contact there was gracious, accommodating and proficient.  (As it turns out, I had reached this person on his last day of work at ICEL and got an extremely fast turnaround - I have never met a faster yet impeccable editor.) The ICEL will again edit, so there may be another round of revisions (ICEL may be a bit more stringent on repeated words and phrases.)  Permission is contingent on how you intend to use the copyrighted texts.  If you plan to sell it, then a contract will be involved; otherwise you should eventually get an email something like this: 

guidance

Thanks to the priests who impact(ed) my spiritual journey: Fathers Stan, Jerry, Charlie, Jim, John, Steve, Michael and Frank.  Especially to Father John, my cousin, and Father Steve, a former technical colleague, who fill in my Church and Latin gaps (the setting was almost named Missa ex Cor - Mass of the Heart Organ.)

Thanks to Dustin Himmerich for a series of emails describing the approval and permission process.