For a single word verb with the same meaning, you can use gestate. From Collins Dictionaries:
- (transitive) to carry (developing young) in the uterus during pregnancy
‘How incredible would it be to gestate a baby and spawn it?.
Times, Sunday Times (2013)’
I should point out that this is a technical-sounding term. In casual conversation it will usually sound tongue-in-cheek, like the example above. This joking sort of usage is how I've mostly used it ("I can't do the dishes, honey, I'm too exhausted from gestating all day"). It would fit in your academic-sounding example, though:
Surrogacy is an arrangement or agreement whereby a woman agrees to gestate (a[n] embryo/fetus/baby/child) for another person or persons, who will become the newborn child's parent(s) after birth.