Applying to a PhD with a weak masters thesis I am working on my masters thesis which is the first time that I have ever really taken part in research activities. Throughout the experience I have learned a lot and I think that I would like to pursue a PhD in the field that I am currently working in right now. However, I feel that when all is said and done with my masters thesis, the end product will not be that great. There are a number of factors that I think contribute to this - the first being that it's my first time doing original research and second I am the first student of my supervisor, so we are experiencing the student-supervisor relationship both for the first time. But given that, I think it is primarily a function of my introduction to research. A similar analogy would be a first year undergraduate student that doesn't do well, but finds their groove in the last couple years of the program. 

My concern, however, is that if I produce a low quality thesis that I might actually be disadvantaged compared to a recently graduated undergraduate student. For example, my undergrad/grad GPA's were (3.5/3.8) and assume that the undergraduate also had a GPA of 3.5-3.8, but I can foresee that a committee might view my weak thesis as an inability to do research in contrast to the undergraduate does not have any published research either, but doesn't show signs of being a "weak researcher" by nature of not having done any. 

So the question really becomes whether or not somebodies MSc experience is a good predictor of how they would fair out as a PhD student, and whether somebody applying to a program with an MSc can actually be disadvantaged. 

Edit: Perhaps mentioning my perspective on the MSc experience in my SOP might be of benefit? Unless, all of this might be moot if entrance committees might not even read your MSc thesis.
First, you may be judging your own work more harshly than is warranted. It may not be as weak as you think. It is fairly common for problem to appear easier after it is solved than before you began. 

Second, no one is likely to read your master's thesis as part of a doctoral application. They will take note of it, and any grade you got, and the general field. At most a few people will read the abstract. They are too busy to read it, frankly. 

Third, an application for doctoral study is normally judged on a wide range of criteria with different candidates scoring higher or lower on the various elements. But the overall picture is what is important. "Does this candidate have the necessary background?" "Does this candidate have a good chance of success now and hereafter?" And since most programs are competitive, "How does this candidate compare with others in the pool?". 

But, if you are accepted then you still have two hurdles. The first is passing comprehensive qualifying exams. Some advanced coursework may still be needed to get you to that point. The other is writing an acceptable thesis under the direction of an advisor, along with whatever research enables that. 

In the US a lot of doctoral students enter the program with no research experience at all. You already have an advantage there, no matter how your recent thesis turned out. And, even at the doctoral level, not every dissertation has the same impact. It represents early work, not the person's best career work. 

Make sure your CV is solid with your background accomplishments. Don't negatively characterize your thesis. Make sure your SoP explains your goals, short and long term, along with how your background supports that. 

And, though you name a specific school, don't cast a narrow net for a doctoral program. Give yourself more than one chance.