Warmth, quiet and darkness are the whole of your first aid here. Your task is narrow and important: warm the bird gradually, disturb it as little as possible, and get professional direction quickly. A chilled dove or an injured garden bird is fragile in ways that are hard to judge from outside, so resist the urge to examine, feed or treat it yourself.
Set up the box before you pick the bird up
Gather a small cardboard box with a lid, a soft cloth without loops or loose threads, a warm-water bottle or a heat pad, a towel to wrap the heat source, and your phone. Punch a few air holes in the box lid or upper sides. Line the base with the cloth, folded smooth so small toes cannot catch. Fill the bottle with warm, not hot, water, wrap it in the towel, and place it against one outside wall of the box so heat passes through the cardboard gradually and the bird can rest nearer to or farther from that warm side. A small box matters: in a large space a weak bird topples and flaps, wasting energy and risking further injury.
Move the bird once, gently
Wash your hands, then approach quietly and lay a light cloth over the bird before lifting, which calms it and protects you from a startled peck. Support its whole body in both hands, keep it upright rather than on its back, and lower it into the box in one smooth motion. Close the lid promptly. Handle the bird this once only. Repeated checking, stroking or lifting chills it again, raises stress, and can turn a survivable injury into a fatal one. Place the box in a warm, dim room, off the floor, shut away from household pets, children and television noise.
Warm gradually, never fast
Gradual is the rule that protects a chilled bird. A bird warmed too quickly, with very hot water or direct heat blown onto it, can deteriorate rather than recover. Check the bottle after about half an hour by touching the outside of the box: the warm side should feel comfortably warm to your hand, never hot. Replace cooling water with fresh warm water as needed, working from outside the box where you can, and open the lid only briefly. If the bird starts panting, holds its wings away from its body or moves restlessly to the far corner, the box is too warm, so remove the heat source and let the temperature ease down slowly.
What never to offer
Do not offer food, and do not drip water into the beak. A cold bird cannot digest food safely, an injured bird may inhale fluid, and the wrong food can cause harm that a rehabilitator then has to undo. Do not try to splint a wing, clean a wound, or give any medicine. Your restraint is not doing nothing: warmth, darkness and quiet are exactly what experienced rehabilitators ask finders to provide first, because those conditions keep the bird stable until skilled hands take over.
Making the call count
Phone a veterinary practice or a wildlife rehabilitator as soon as the box is settled, and describe where you found the bird, what you can see without handling it, and how long it has been boxed. Ask where to take it and whether they want you to bring it in or wait for collection, since many rehabilitators work from home addresses that are not public. Transport the closed box in a quiet car, keep it level, and hand over your notes with the bird. Knowing where the nearest help sits matters beforehand too, and KinsLe's Pet Services Directory lists practices across your area for exactly such moments.