A double lumen catheter has an elongated tube with a proximal first cylindrical portion having an internal divider to define a pair of discrete lumens including a shorter lumen opening at approximately the distal end of the first cylindrical portion, and a longer lumen opening at the distal end of the elongated tube. The longer lumen is further defined by a distal second cylindrical portion of the elongated tube which has a diameter substantially less than the diameter of the first proximal cylindrical portion of the elongated tube. To promote insertion and to perform a dilator function, the distal end of the tube has a smooth conical tapered tip that smoothly merges with the distal second cylindrical portion of the elongated tube. Preferably, the lumens are "D" shaped in the first cylindrical portion, a plurality of outlet holes are provided in the vicinity of the conical tapered tip, and the shorter lumen opens at an aperture having a bevel. The double lumen catheter is particularly advantageous when a tunneling procedure or blind technique must be used, for example, to reach a vein under the collar bone or neck.