We were in flight in a C-130 (military) aircraft at approximately 30 thousand feet, and at least 500 miles south of Louisiana over the Gulf of Mexico. We were on a navigator training mission in clear skies on a very sunny day. We were outside of US radar control range so no immediate reporting method was available. I was in the co-pilot's seat, and I was the first to notice the object which appeared to hover at a constant distance just above and in front of us. It didn't appear to be any threat, and just looked like a big silver pinball floating in front of us. At first, I thought it might be possible that we had sighted a weather balloon and I mentioned this to the rest of the crew. But as we continued southbound at approximately 300 knots, and this object maintained perfect position in front of us, it became obvious that this was no weather balloon. Other crew members also mentioned that a weather balloon is usually of an elongated shape, and this object appeared perfectly round. There were no visible windows or appendages on this object, but it appeared to be under intelligent control (maintaining perfect speed and altitude to match ours). Our entire crew came forward next to my seat (one by one) and took turns viewing this object through the front windscreen. None of our crew members had a camera with them on this day, and we did have a long discussion about the lack of proof should we decide to report this sighting. Since we were on a navigator training mission, we were obligated to maintain a steady course (approximately southbound) and did so until we reached the planned turn around point. We continued on course for approximately one hour, checking the position of the object every few seconds. It NEVER changed position, relative to our aircraft, for at least an hour. Since this object didn't appear threatening in any way, we eventually stopped even discussing it. The object disappeared somehow between my frequent checking of it's position, and I didn't see it depart our flight path. It was there one second, and gone the next. My crosscheck for airborne traffic as a copilot had always been very thorough, so I remember being surprised by the fact that the object was gone without my seeing it leave. The rest of the crew (except the pilot) seemed to forget all about the object after fifteen or twenty minutes. We discussed this sighting on the trip back, and decided not to report it at that time.