The wording was delivered of was used in the official announcement at Buckingham Palace (image from the Press Association via BBC News):
OED has
3 a. To disburden (a woman) of the fœtus, to bring to childbirth; in pass., to give birth to a child or offspring. Rarely said of beasts. (The active is late and chiefly in obstetrical use.)
which is a use of deliver in an older [and obsolete] sense meaning relieve:
†2. a. To free, rid, divest, clear (a) of, (b) from.
The language is formal, and perhaps a little archaic.
