First, begin any essay assignment with clear instructions on the amount of outside assistance that is allowed.
Many universities have "drop in writing tutors" who are paid (albeit often rather modestly) to help students with their writing assignments. These tutors are well-trained on how not to help too much: e.g. they probably do not write down as much as a complete sentence that the student has not already suggested.
Having students expressly acknowledge the outside help that they get, even if absolutely legitimate, helps towards fostering good academic practices.
Thus if they go to the university's writing center or some similar entity, that's fine (assuming that above it has been clarified by the instructor to be fine; but if your university has a writing center then it seems like a bad idea not to allow your students to use it in at least some ways), but they should still acknowledge it. This goes a long way to combating the problem of "mysterious diction", i.e., the isolated turn of phrase that you just don't quite believe came from the student.
You should really enforce this practice: if it turns out that a student went to the writing center and didn't report it, lower their grade at least slightly: this helps to reiterate that this is very important to you. In fact:
Communicate absolutely clearly that in academia there is no sin worse than passing off someone's ideas and/or work as your own.
I think that most undergraduates know that this is something they are not supposed to do, but many don't seem to understand that this is a very serious crime. To me at least, the crime is so serious that -- in light of the difficulties of definitively "catching" the student -- the penalties should be clarified in advance to be quite severe. At my university, the absolute minimum penalty for this is a grade of 0 on the assignment. I would say though that a student who paid someone else to write a paper for them and is caught deserves to fail the entire course. Make this clear.
Structure a written assignment more than just asking for a full-blown paper at the end.
This has the merit of being a good practice for other reasons. Even in mathematics courses I sometimes ask students to write papers, and I have learned from hard experience that if you don't check in with them while they are doing the work then the final quality of the product is going to be ridiculously spotty: e.g. it will turn out that some students simply didn't understand what you were asking them to do to the extent that you do not recognize what they turn in as being a specimen (however poor) of what you asked for.
For a significant paper in an undergraduate course it could be appropriate to meet with the student several times before they turn in the final version: once to determine and have you sign off on an appropriate topic, and at least once to show you their partial work, giving you a chance to adjust them back on track if need be.
If you break the assignment up into pieces like this, then it becomes at the same time harder for a student to pay someone else to do it and also easier for them to do it themselves. I have to imagine that a lot of this professional paper-writing is done when a student realizes that they have waited until the last 24-72 hours before the paper is due. Someone can write a good paper under those severe time constraints, but they can't.
Consider having a classroom component accompany any significant written work.
This works better in some courses than others, depending on both the size of the course and the subject matter. Hearing 25 different presentations on whether and in what way Madame Bovary was a proto-feminist sounds like a rather dreary use of class time. But let me say that in every graduate (math) course I teach, whether I assign written homework or not I always have a "problem session", usually about once a week and lasting at least an hour, in which the students present their solutions to me and to each other (with whatever written notes they want, of course). I don't do this because I'm worried that their written work will not be their own, not really, but it is definitely a different thing to cobble together a solution from two different texts and a website (this is a totally legitimate thing to do, in my courses: in fact this sounds like a reasonable description of some of my own research!) than to actually present it in front of other people. (Again, I've learned this from hard experience: in many of the courses I teach, the course text is my own notes, sometimes written on the fly and sometimes already written. It is remarkable to me how I can give a better or worse rendition of something that I looked up from my own lecture notes on any given day. Sometimes I don't succeed in getting the material as firmly in my head as I would like in the time allotted to prepare that day's lecture.)
If there is both an oral and a written component, then layering is again a good thing: maybe the student turns in some preliminary written thing, then they give a presentation on it, then they improve their written work based on the feedback they got from the presentation. Again, this seems more appropriate in some courses than others, but when it is appropriate the students learn a lot: perhaps they will even be too busy learning and working to think to get someone else to write their paper.