An analytical and numerical study of the linear Saffman-Taylor instability for a Maxwell viscoelastic fluid is presented. Results obtained in a rectangular Hele-Shaw cell are complemented by experiments in a circular cell corroborating the universality of our main result: The base flow becomes unstable and the propagating disturbances develop into crack-like features. The full hydrodynamics equations in a regime where viscoelasticity dominates show that perturbations to the pressure remain Laplacian. Darcy's law is expressed as an infinite series in the cell thickness. An unique dimensionless parameter (lambda) over bar, equivalent to a relaxation time, controls the growth rate of the perturbation. (lambda) over bar depends on the applied gradient of pressure, the surface tension, the cell thickness, and the elastic modulus of the fluid. For small values of (lambda) over bar Newtonian behavior dominates whereas for higher values of (lambda) over bar viscoelastic effects appear. For the critical value (lambda) over bar = (lambda(c)) over bar similar or equal to 10 a blowup is predicted and fracture-like patterns are observed. (C) 2012 Elsevier B.V. All rights reserved.