Potential Game-Changer?

As my devoted followers will recall, I recently had an intership interview and was then left with making the decision of whether or not to take the unpaid position. I had ultimately come to the conclusion that I couldn't logistically or practically take an unpaid position, especially one that wasn't necessarily directing me towards the career path I desired, despite its potential to offer valuable experience and maybe a full-time job at the end of the internship.

Today, I knew I was going to have to call the company and inform them of the decision. I explained to them my reasons for turning down their offer, but instead of the phone call ending there, I was instead faced with another decision. They asked me if my decision would be different if they were able to offer me a stipend and how much would I need? As you can imagine, this certainly caused me to reconsider.

So now, I am again faced with yet another decision. Obviously, pending the answer of certain questions such as how much the stipend will be, what sort of daily commitment would this internship be and how long would the duration of the internship be, I could potentially see this as a viable offer. At the same time, do I hold out in hopes of another full-time position, considering I am only a month into my job hunt? I graduated early to get a head start on the job hunt and not have to be mixed into the search with a sea of other graduates, so do I stop the search now or persevere?