Replication inside a living cell, carried out by DNA polymerase, has an error rate far below that predicted by equilibrium thermodynamics from the affinities between nucleotides and a polymerase complex. The high fidelity is achieved through several distinctly different molecular mechanisms that include a nucleotide insertion checkpoint and 3'-5' exonuclease activity. The checkpoint mechanism has recently been articulated as a new paradigm for high specificity. A rigorous thermodynamic analysis of the bare DNA polymerization reaction, i.e., in the absence of exonuclease activity and proofreading, is developed in this paper. The analysis (a) reveals the important role of nonequilibrium steady-state (NESS) flux that drives high fidelity, (b) quantifies the error rate of the polymerization reaction as a function of free energy input through sustained non-equilibrium between chemical species, (c) bridges the 'thermodynamic' and 'kinetic' views of specificity and (d) generalizes the theory of kinetic checkpoints and provides it with a sound thermodynamic basis. The underlying mechanism again calls attention to the energy expenditure in heightened biomolecular specificity, a concept first developed by Hopfield and Ninio in the mid-1970s. The mechanism discussed in the present paper is not limited to DNA replication alone; it may be applicable to other biochemical systems.