Tap the mortar around the brick with a mason's bolster, which is a tool that looks like a wide chisel. When the mortar starts to crumble, switch to a larger bolster, and tap out a larger hole around the brick. Be careful not to damage or scrape the other bricks with the bolster. Damaging the other bricks could cause them to spall at a later date.
Tug on the spalled brick gently. If it won't come out of the hole, continue to tap around the brick with a bolster until it will.
Line the space where the brick used to be with mortar. Allow the mortar to sit for 10 minutes.
Insert the new brick into the space and fill in any holes in the mortar with a narrow pointing trowel.