A stentless support structure capable of being at least partly assembled in situ. The support structure comprises a braided tube that is very flexible and, when elongated, becomes very long and very small in diameter, thereby being capable of placement within a small diameter catheter. The support structure is preferably constructed of one or more thin strands of a super-elastic or shape memory material such as Nitinol. When released from the catheter, the support structure folds itself into a longitudinally compact configuration. The support structure thus gains significant strength as the number of folds increase. This radial strength obviates the need for a support stent. The support structure may include attachment points for a prosthetic valve.