Having bought and then sold the same BMW on Bring-a-Trailer (BaT) myself over the past few years, I can also recommend them for the sale of your car. But, there are a number of caveats to consider, too. The typical auction on BaT runs for 7 days, and during that time any interested party, whether a potential buyer or simply an enthusiastic follower, can post any number of questions and comments about the car, whether constructive or not. Sellers are highly encouraged to stay closely connected to their auction during the week of activitiy, and posting honest, complete and timely replies to questions is very important. As with any older classic car, Volvo 1800's have quite a bit of history, folklore, and facts that come with them, and unless you are well-informed about those issues, you can get over your head in a hurry. In any case, honesty and full disclosure is paramount. Having at least a semi-expert in 1800's at the ready to help you out would be a real plus.
You will also have to post quite a few digital images of the car, of reasonably high quality, taken at every conceivable angle, including any visible defects on the car. A few digital videos of the car sitting, and driving, are also highly recommended. The BaT website discusses all of this in detail, and they even have links to professional photographers who can be hired for this purpose. Don't underestimate the time, and possible expense, of prepping your car for auction. The car may look gorgeous sitting in the driveway, but if it has sat motionless for an extended period of time, it may well need a fair amount of refurbishing (fresh fluids, tune-up, new tires, spot rust remediation, etc.) before it is ready to sell. Make sure you have all records available, and definitely a clear title of ownership.
I would suggest registering with BaT (it's free, but you need to post a credit card #), after which you can access all the features of their site. Most important will be looking into the history of past 1800 auctions, and there have been many. You will quickly see how best to prepare your own auction. There are headaches involved, but they probably give your car the greatest exposure to informed and enthusiastic potential new owners, and that's what you really want. Good luck!