I dont know anything about the transition, but I would strongly suggest talking to someone at the school you are considering and telling them you goals.

in my mind most ppl who enter a "design masters" program, have sum sort of background in design, ie fashion, art, architecture, landscape arch, etc. or construction, and the program typically will want to see a portfolio, and because of your background u'd better have a serious portfolio. (i.e. no stick men)

that said, what will your thesis explore / prove?

an alternative wud probably to get a masters in art, and then in architecture if your terminal goal is to get licensed.
 

Read:    Architecture?  by Roger Lewis.  Good overall description of life in the profession.  You need to make sure you really want to be an architect before you decide on schools.  Just my opinion.

Sounds like you need a Master of Architecture program.  They are commonly referred to as an M. Arch 1 or "first professional degree"  They often have some prerequisites like calculus, physics, a drawing class, etc. that you may or may not have done in undergrad.  If you don't have them, consider taking them at a community college.  Find a few schools you are interested in and see what they require for admission.  The tuition costs, financial aid, and level of selectivity varies greatly from school to school. 

I've known a fair number of students in M.Arch programs that had decidedly non-technical undergrad degrees.

Essentially what Thisisnotmyname said.

What you are really looking for is a first professional degree Master of Architecture program with no architectural background requirements. Which will be a 3+ year program.

Which is to clarify that:
 - "first professional degree" also refers to Master of Architecture programs intended for people with previous architectural experience ( which is a 2 year or less program)

- The names for theses two degrees is at the discretion of the schools and thus an MArch 1 (while often the program you are looking for) is not always referring to the 3+ year program for non-arch backgrounds.

You're in a firm - so make friends with the design staff, keep your eyes and ears open.  Ask for recommendations on books to read, check out the magazines that they subscribe to, go to lectures with them, maybe crash a lunch-and-learn.  Schools are more-or-less academic and the day-to-day may be less interesting, but there are a lot of things you won't learn in school that you can in a firm.

Making friends with the staff: ask about how they got into the field, where they went to school and why, what keeps them going in an often grueling profession.  Even though you're admin, see if they're willing to let you take on some tasks.  I always need an extra set of eyes to catch spelling and grammar, even just taking a look at a drawing to see if it makes sense to a lay person (most clients are lay people, so this is helpful for presentation stuff).

Getting prepped for entering the field properly: see what programs are around (in NYC: Columbia, CCNY, Pratt, Cooper Union, Parsons/New School) and get to know some students and/or faculty (ask coworkers), what they're requirements and what they want in their students.  Portfolio's are a big part, though less so for non-architecture people entering a Masters.  Its usually to show creative potential, so taking drawing, painting, sculpture, or photography courses are great (I went a similar path and a plant drawing course really helped my portfolio as well as my interest in photography).

A lot of schools focus on people w/o backgrounds in architecture and most schools will offer a degree in that track.  Typically the first year is with those in a similar position and starting in the second they throw you in with those that have undergraduate architecture degrees.  In my school we had people entering second careers (with first careers ranging from lawyers to science teachers to editors), and it was a great experience as each had their own reasons for wanting to do architecture and it made for great conversations.

Quite frankly, as others have pointed out, you will need an M.Arch to really get into the field unless you have some crazy skill set.  But if you plan ahead it's not too painful.