The usual name for a “Bible service” today is a “Celebration of the Word of God”.
In the 2007 Apostolic Exhortation Sacramentum Caritatis, Pope Benedict encouraged their use in particular circumstances:
“50. … In this regard, I would like to call attention to a pastoral problem frequently encountered nowadays. I am referring to the fact that on certain occasions – for example, wedding Masses, funerals and the like – in addition to practising Catholics there may be others present who have long since ceased to attend Mass or are living in a situation which does not permit them to receive the sacraments. At other times members of other Christian confessions and even other religions may be present. Similar situations can occur in churches that are frequently visited, especially in tourist areas. In these cases, there is a need to find a brief and clear way to remind those present of the meaning of sacramental communion and the conditions required for its reception. Wherever circumstances make it impossible to ensure that the meaning of the Eucharist is duly appreciated, the appropriateness of replacing the celebration of the Mass with a celebration of the word of God should be considered.”
Nearly all the Church’s ceremonies can be celebrated outside of Mass and the liturgical books describe how this can be done. The Ceremonial of Bishops has a detailed description of how the Bishop presides at one. The RCIA liturigcal book encourages the use of them for catechesis. The Book of Blessings has lots of blessings which can be done as part of a Celebration of the Word of God.
The 2001 Directory on Popular Piety and the Liturgy refers to your quote:
“*Prayerfully Hearing the Word of God *
193. The Council’s call for the “sacred celebration of the word of God” at significant moments throughout the Liturgical Year (footnote 227 Cf. Sacrosanctum Concilium 35, 4.), can easily find useful application in devotional exercises made in honour of the Mother of the Word Incarnate. This corresponds perfectly with the orientation of Christian piety(228) and reflects the conviction that it is already a worthy way to honour the Blessed Virgin Mary, since it involves acting as she did in relation to the Word of God. She lovingly accepted the Word and treasured it in her heart, meditated on it in her mind and spread it with her lips. She faithfully put it into practise and modelled her life on it(229). 194. “Celebrations of the Word, because of their thematic and structural content, offer many elements of worship which are at the same time genuine expressions of devotion and opportunities for a systematic catechesis on the Blessed Virgin Mary. Experience, however, proves that celebrations of the Word should not assume a predominantly intellectual or didactic character. Through hymns, prayers, and participation of the faithful they should allow for simple and familiar expressions of popular piety which speak directly to the hearts of the faithful”(230).”
Footnotes:
(228) Cf. PAUL VI, Apostolic Exhortation Marialis Cultus, 30.
(229) Cf. ibid., 17; Collectio missarum de beata Virginis Mariae, Praenotanda ad lectionarium, 10.
(230) CONGREGATION FOR DIVINE WORSHIP, Circular Letter Guidelines and proposals for the celebration of the Marian Year, 10.
More information on them is available from the book “Documents on the Liturgy 1963-1979”, Liturgical Press, Minnesota, 1982, ISBN 0-8146-1281-4. Look up “celebrations of the word” in the index.