It is proposed that the randomly diluted elastic network of Hooke's-law central-force springs in two dimensions may have a hitherto unrecognized splay-rigid phase in which the bulk and shear moduli vanish but the Frank elastic constant, K, is nonzero. Exponents for the associated splay-rigidity percolation and torsional resistance are estimated by series-expansion techniques. These exponents are related to that for K through the node-link picture.