The lensed double QSO 0957+561 has a well-measured time delay and hence is useful for a global determination of H _ { 0 } .
Uncertainty in the mass distribution of the lens is the largest source of uncertainty in the derived H _ { 0 } .
We investigate the range of H _ { 0 } produced by a set of lens models intended to mimic the full range of astrophysically plausible mass distributions , using as constraints the numerous multiply-imaged sources which have been detected .
We obtain the first adequate fit to all the observations , but only if we include effects from the galaxy cluster beyond a constant local magnification and shear .
Both the lens galaxy and the surrounding cluster must depart from circular symmetry as well .

Lens models which are consistent with observations to 95 % CL indicate H _ { 0 } = 104 ^ { +31 } _ { -23 } ( 1 - { \bar { \kappa } _ { 30 ^ { \prime \prime } } } ) { km } { s } ^ { -1 } { Mpc } ^ { -1 } .
Previous weak lensing measurements constrain the mean mass density within 30 \arcsec of G1 to be { \bar { \kappa } _ { 30 ^ { \prime \prime } } } = 0.26 \pm 0.16 ( 95 % CL ) , implying H _ { 0 } = 77 ^ { +29 } _ { -24 } { km } { s } ^ { -1 } { Mpc } ^ { -1 } ( 95 % CL ) .
The best-fitting models span the range 65–80 { km } { s } ^ { -1 } { Mpc } ^ { -1 } .
Further observations will shrink the confidence interval for both the mass model and { \bar { \kappa } _ { 30 ^ { \prime \prime } } } .

The range of H _ { 0 } allowed by the full gamut of our lens models is substantially larger than that implied by limiting consideration to simple power law density profiles .
We therefore caution against use of simple isothermal or power-law mass models in the derivation of H _ { 0 } from other time-delay systems .
High-S/N imaging of multiple or extended lensed features will greatly reduce the H _ { 0 } uncertainties when fitting complex models to time-delay lenses .