We have fabricated longitudinal nanoconstrictions in the charge-density wave conductor (CDW) NbSe3 using a focused ion beam and using a mechanically controlled break-junction technique. Conductance peaks are observed below the TP1=145K and TP2=59K CDW transitions, which correspond closely with previous values of the full CDW gaps 2Δ1 and 2Δ2 obtained from photoemission. These results can be explained by assuming CDW-CDW tunneling in the presence of an energy gap corrugation 2 comparable to Δ2, which eliminates expected peaks at ±|Δ1+Δ2|. The nanometer length scales our experiments imply indicate that an alternative explanation based on tunneling through back-to-back CDW-normal-conductor junctions is unlikely. © 2006 The American Physical Society.