Most of the other answers have covered your options, but there is one more possibility you might want to consider before buying an electric guitar. They make sound-hole dampeners for acoustic guitars that are used to fight feedback when playing an acoustic amplified on stage, but they also provide some dampening of the volume of the guitar. Check the link below for one such device. Again, this isn't exactly made for your purposes, but it would help. It won't make the guitar silent, but it would reduce the overall volume a bit. Stuffing the soundhole as someone else mentioned would also go a long way towards dampening your overall volume. A t-shirt or two can be pushed inside without much effort.
I can tell you that when I play acoustic guitar in my apartment, especially late at night, I put the pick away and just strum lightly with my thumb/fingers to make the least amount of noise possible. Doing this results in a pretty quiet overall volume, and I'm pretty sure no one outside of my apartment can hear me playing. With a light touch it is possible to play an acoustic guitar very quietly, it's really just up to how hard you strum/pluck the strings. Of course I do have more experience and have been playing for many years, so it might not be as easy for a beginner to modulate the volume like I do but it shouldn't be very hard to learn how to do so.
I would reiterate that the quietest solution (albeit probably the most expensive), might be an electric guitar. It does make some small amount of noise, mostly from the picking of the strings, especially if you are strumming forcefully, but again this can be controlled with a careful right hand. They make tons of headphone amps for electric guitars, but you can also just play one without an amp and it's loud enough for you to hear what you are playing but not so loud that anyone outside your room would notice.