Your Model 1865 Spencer Carbine is chambered for 56-50 rimfire ammo. That round hasn't been loaded in almost a century and probably wouldn't go off if you found any. Most of us convert to centerfire by replacing the rimfire breechblock with a modern made conversion block. If you reload, cases, bullet molds and dies are available, or you can cut down 50-70 cases. There are a couple places loading centerfire 56-50 ammo, but it is usually made for the Italian reproductions, so it won't always shoot well in originals (due to bore size variation) or extract reliably. Search through the old posts here and you'll find a wealth of information on the centerfire conversion. Be sure you have your Spencer checked by a gunsmith that specializes in antiques before shooting it.
Your serial number is unusually low for an M-1865 carbine. Most of the first 2000 are rifles. Your carbine would have been delivered around the end of March, 1865. It is generally believed that M-1865s didn't make it to the field before Lee's surrender in April. However, the M-1865s were very much anticipated, so the first ones may have been issued in late March. They did see service in Texas during the buildup in mid-1865 to prevent a southern invasion via Mexico, and in the US regular cavalry until being replaced by Trapdoor Springfield carbines in 1873-'75. A lot of them were sold to France for the Franco-Prussian War, and found their way back to the US afterward.
Spencers are great fun to shoot and surprisingly accurate with the right ammo, and within about 200 yards. Enjoy your piece of history!