We present a 30 h integration Very Large Telescope X-shooter spectrum of the Lyman series forest towards the z = 7.084 quasar ULAS J1120+0641. The only detected transmission at S/N > 5 is confined to seven narrow spikes in the Lyα forest, over the redshift range 5.858 <z < 6.122, just longward of the wavelength of the onset of the Lyβ forest. There is also a possible detection of one further unresolved spike in the Lyβ forest at z = 6.854, with S/N = 4.5. We also present revised Hubble Space Telescope F814W photometry of the source. The summed flux from the transmission spikes is in agreement with the F814W photometry, so all the transmission in the Lyman series forest may have been detected. There is a Gunn-Peterson (GP) trough in the Lyα forest from z = 6.122 all the way to the quasar near zone at z = 7.04. The trough, of comoving length 240 h-1 Mpc, is over twice as long as the next longest known GP trough. We combine the spectroscopic and photometric results to constrain the evolution of the Lyα effective optical depth (τGPeff) with redshift, extending a similar analysis by Simpson et al. We find τGPeff ∝ (1 + z)ξ where ξ = 11.2+ 0.4-0.6, for z > 5.5. The data nevertheless provide only a weak limit on the volume-weighted intergalactic medium (IGM) hydrogen neutral fraction at z ~ 6.5, xH i > 10-4, similar to limits at redshift z ~ 6 from less distant quasars. The new observations cannot extend measurements of the neutral fraction of the IGM to higher values because absorption in the Lyα forest is already saturated near z ~ 6. For higher neutral fractions, other methods such as measuring the red damping wing of the IGM will be required.