The final antiphon at all of the hours of the classical Roman Breviary in the Octave of Easter, piously recited three times a day instead of the Angelus until Pentecost.

The beautiful translation given in a devout book ("Common Prayers" pub. Society of SS. Peter and Paul in 1922) is:

Joy to thee, O Queen of Heaven! Alleluia!
He whom thou wast meet to bear: Alleluia.
As he promised hath arisen: Alleluia.
Pour for us to God thy prayer: Alleluia.

V. Rejoice and be glad, O Virgin Mary: Alleluia.
R. For the Lord hath risen indeed: Alleluia.

Let us pray.

O God, who by the resurrection of thy Son our Lord Jesus Christ hast given joy unto the world: grant we beseech thee; that through his mother the Virgin Mary we may obtain the joys of everlasting life. Amen.

This is the translation I use devoutly and find nothing in it wanting, but am eager to restore my decayed Latinity prior to going to University as a mature student after a long illness so am beginning to produce my own word-by-word and have struck a few curiosities I am sure you can clear up.

The Latin text is: 
Regina Coeli, laetare, alleluia,
Quia quem meruisti portare, alleluia,
Resurrexit, sicut dixit, alleluia,
Ora pro nobis Deum, alleluia.

V. Gaude et laetare Virgo Maria, alleluia
R. Quia surrexit Dominus vere, alleluia

Oremus

Deus, qui per resurrectionem Filii tui, Domini nostri Jesu Christi, mundum laetificare dignatus es: praesta, quaesumus; ut per eius Genetricem Virginem Mariam, perpetuae capiamus vitae aeternae. Per eundem Christum Dominum nostrum. Amen.

And my own translation

O Queen of Heaven, rejoice, alleluia,
For (He) whom thou hast merited to bear, alleluia,
Hath arisen, as He hath said, alleluia,
Pray for us to* God, alleluia.

V. O Rejoice and be thou glad, O Virgin Mary, alleluia
R. For the Lord hath risen indeed, alleluia

Let us pray.

O God, who by the Resurrection of Thy Son, Our Lord Jesus Christ, hast been pleased to gladden the world*: grant (Stelten), we beseech thee, that by his Mother the Virgin Mary we may obtain (Stelten, capio, capere, cepi, captum is "to seize, to take hold of" more usually) the joys of everlasting life.

The difficulty I seem to have is with "mundum". I presume "Deum" in the antiphon itself has a laconically omitted "ad" requiring the accusative (ad governing the accusative, to not unto). Am I correct, following HPV Nunn, that mundum is the subject of laetificare as a verb (extrem. lit. hast been pleased the world to gladden) (HPV Nunn, An introduction to Eccl. Latin, p. 50) and is in the accusative for that reason - as the subject of an infinitive per Nunn? 

The dative construction in English translations would then simply be a requirement of "give joy to" rather than "gladden". 

Do please forgive the almost inevitable blunders - I am at the very beginning of trying to learn the Latin of the Breviary and the Missal after six years away from any study.