Cross-stream migration and stable orientations of elliptic particles falling in an Oldroyd-B fluid in a channel are studied. We show that the normal component of the extra stress on a rigid body vanishes; lateral forces and torques are determined by the pressure. Inertia turns the long side of the ellipse across the stream and elasticity turns it along the stream; tilted off-centre falling is unstable. There are two critical numbers: the elasticity and Mach numbers. When the elasticity number is smaller than critical the fluid is essentially Newtonian with broadside-on falling at the centreline of the channel. For larger elasticity numbers the settling turns the long side of the particle along the stream in the channel centre for all velocities below a critical one, identified with a critical Mach number of order one. For larger Mach numbers the ellipse flips into broadside-on falling again. The critical numbers are functions of the channel blockage ratio, the particle aspect ratio and the retardation/relaxation time ratio of the fluid. Two ellipses falling near to each other, attract, line-up vertically and straighten-out with long sides vertical. Stable, off-centre tilting is found for ellipses falling in shear-thinning fluids and for cylinders with flat ends in which particles tend to align their longest diameter with gravity.