Note that such a connection makes an integrator, of some limited DC gain (due to the resistor), and some cutoff frequency (due to the capacitor).
That cutoff (GBW), can go up to a limiting value of the amp's GBW, when C = 0. Or put another way, we can calculate an effective capacitance from the amp's GBW, and say that, even for in-circuit C = 0, there is some minimum actual value of C always present, and we can only connect more capacitance in parallel with it to further lower GBW.
Note that this is really only true at low frequencies, or large capacitance, so that the opamp still has high loop gain at frequencies of interest. For frequencies near GBW, there's all the non-idealities of the circuit: the input and output impedances rise, phase shifts, there's even a feed-forward path directly through the capacitor so gain actually goes to zero and reverses (goes positive) past some point!
And yeah, as for stability, that depends on more subtle things; most importantly, how much phase shift is incurred around fT. Note that, especially as you go up in bandwidth, you may find a lot of "decompensated" amps, that are only stable down to a certain noise gain -- that is, a measure of how much negative feedback is applied around the amp. These will NOT be stable with a unity-gain configuration (note that, asymptotically speaking, an integrator requires a unity-gain-stable amp: at high enough frequencies, the capacitor looks like a short circuit, so the amp looks like a voltage follower with +in = GND). If you must, you can sometimes improve stability, by reducing noise gain at high frequencies -- a typical way of doing this is placing a cap between input pins (+in to -in). This effects a lower input voltage difference, at frequencies where the amp's gain is low and therefore the input voltage would otherwise be rising; which in turn reduces the amount of negative feedback applied, which also *drumroll* increases the noise level, hence the term "noise gain". (By definition, noise gain is the expected output noise or offset voltage, given the amp's input voltage offset or noise level. Note that input-referred noise is nothing more than an AC offset voltage.)
Tim