The replication of nucleic acids is central to the origin of life. On the early Earth, suitable non-equilibrium boundary conditions would have been required to surmount the effects of thermodynamic equilibrium such as the dilution and degradation of oligonucleotides. One particularly intractable experimental finding is that short genetic polymers replicate faster and outcompete longer ones, which leads to ever shorter sequences and the loss of genetic information. Here we show that a heat flux across an open pore in submerged rock concentrates replicating oligonucleotides from a constant feeding flow and selects for longer strands. Our experiments utilize the interplay of molecular thermophoresis and laminar convection, the latter driving strand separation and exponential replication. Strands of 75 nucleotides survive whereas strands half as long die out, which inverts the above dilemma of the survival of the shortest. The combined feeding, thermal cycling and positive length selection opens the door for a stable molecular evolution in the long-term microhabitat of heated porous rock.