Stimme ich voll zu! Diese Bedeutung (christliches Werk, christlicher Dienst) habe ich aus den Wöerterbucheinträgen nicht entnehmen können...

Der OP und #1 lassen nicht erkennen, dass der Eintrag bei LEO etwa falsch wäre.
Möglicherweise ist er zu ergänzen; die bisherige Übersetzung halte ich aber für zutreffend.
Vgl. dazu
http://www.ministryformation.org.uk/

Yes, the literal meaning of the word is simply service, and I would certainly support adding additional senses in that broad meaning.

But the more specific sense meaning service as ordained clergy is certainly not wrong.

It might be useful to note that it's sometimes used with 'the,' in contrast to other types of ministry done by lay people.

For example, the more established evangelical churches also have seminaries that ordain candidates for the ministry. Evangelicals may choose to go into the ministry as a career or second career -- that is, become clergy.

Hier ein Fall von "ministry" aus den USA, wo "ministry" eher als"Predigtamt" (eigentlich "Predigtunternehmen" im Bereich "spiritual entertainment") denn als "geistliches Amt" oder "Missionswerk" zu deuten wäre.

http://en.wikipedia.org/wiki/Joyce_Meyer#Ministry

--- ministry
(ˈmɪnɪstrɪ)
n, pl -tries
1. (Ecclesiastical Terms)
a. the profession or duties of a minister of religion
b. the performance of these duties
2. ministers of religion or government ministers considered collectively
3. the tenure of a minister

http://www.thefreedictionary.com/ministry

--- In his examination of the historical use of the terms Predigtamt and Amt (and the Latin terms associated with them) in the Augsburg Confession, Prof. Brug determines that the Predigtamt is not the office of the pastoral ministry, but the act of preaching the gospel.

http://www.wlsessays.net/node/351

1.a. Beruf oder Aufgaben eines Pastors
b. geistliches Amt, Pfarramt, Dienstamt
2. Geistlichkeit; ...
3. Dienstzeit, Amtszeit, (Fest-)Anstellung eines
Pastors

LEO:
ministry [rel.] - geistliches Amt
ministry [rel.] - das Predigtamt
the ministry [rel.] - der geistliche Stand
the ministry [rel.] - die Geistlichkeit kein Pl.
the ministry [rel.] - das Priestertum
ministry to the unemployed [rel.] - die Arbeitslosenbetreuung (kirchlich)
Dictionary: ministry


AHD 5th ed.:
ministry - 1. the act of serving; ministration. 2a. The profession and services of a minister. b. The clergy. c. The period of service of a minister. ...

Oxford Amer. Dict. & Thes.:
ministry - ... 3: the work or office of a minister of religion.

Random House unabridged:
ministry - 1. the service, functions, or profession of a minister of religion. 2. the body or class of ministers of religion; clergy. ... 7. the term of office of a minister. 8. the act or an instance of ministering; ministration; service. ...

Webster's 3rd unabridged:
ministry - 1: the action of ministering: the performance of any service or function for another: MINISTRATION ... 2: the office, duties, or functions of a minister of religion ... 4: the body of ordained ministers of religion: CLERGY ... 6 a: the period of service of a minister of religion b: the total life of service of a religious figure ...

It has been commonly recognized that the highest call for any Christian is the call into "full-time Christian service," that is, the ordained ministry. ... The problem with all of this is that it is unbiblical and certainly not consitent with the model established by Jesus. ...
It was during the Christendom period that a hierarchical structure developed in the Christian church. The kleros (clergy) became the ones who ministered, and the laos (laity) became those to whom they ministered. ...
We know how to bring people into ministries of the congregation--teaching, singing, ushering, serving on boards and committees-- ... But we don't know how to sent people out beyond the boundaries of the congregation to minister in and to the world. ...
This book deals with three crucial needs for ministry in daily life: Christians must be affirmed for their ministries, equipped to better carry them out, and supported in various ways by the congregation.
—William E. Diehl, Ministry in Daily Life: A practical guide for congregations (Alban Institute, 1996), pp. 5-9.

related discussion: street ministry/bus ministry
related discussion: campus ministry
related discussion: bus ministry
related discussion: recovery ministry
related discussion: lay ministry
related discussion: educational ministry
related discussion: teaching ministry

Hmm.

I was assuming that Predigtamt was probably roughly synonymous with 'geistliches Amt,' but what I meant was really the latter, or whatever just means clergy.

I don't actually think that the word 'ministry' literally means preaching as such, though preaching is one of many ministries that Christians, lay or ordained, can undertake.

Perhaps it would be helpful to be sure LEO has entries for both kinds of ministry, e.g.,

ordained ministry
lay ministry
ministry of the laity

It might even be worth searching on 'preaching ministry' -- that's certainly a possible collocation.