We introduce a new form of the Segal–Bargmann transform for a connected Lie group K of compact type. We show that the heat kernel (ρt(x))t>0,x∈K has a space-time analytic continuation to a holomorphic function(ρC(τ,z))Reτ>0,z∈KC, where KC is the complexification of K. The new transform is defined by the integral(Bτf)(z)=∫KρC(τ,zk−1)f(k)dk,z∈KC. If s>0 and τ∈D(s,s) (the disk of radius s centered at s), this integral defines a holomorphic function on KC for each f∈L2(K,ρs). We construct a heat kernel density μs,τ on KC such that, for all s,τ as above, Bs,τ:=Bτ|L2(K,ρs) is an isometric isomorphism from L2(K,ρs) onto the space of holomorphic functions in L2(KC,μs,τ). When τ=t=s, the transform Bt,t coincides with the one introduced by the second author for compact groups and extended by the first author to groups of compact type. When τ=t∈(0,2s), the transform Bs,t coincides with the one introduced by the first two authors.