Context : Quantitative spectroscopy of luminous BA-type supergiants offers a high potential for modern astrophysics .
Detailed studies allow the evolution of massive stars and galactochemical evolution and permits the cosmic distance scale to be constrained observationally .

Aims : A detailed and comprehensive understanding of the atmospheres of BA-type supergiants is required in order to use this potential properly .
The degree to which we can rely on quantitative studies of this class of stars as a whole depends on the quality of the analyses for benchmark objects .
We constrain the basic atmospheric parameters and fundamental stellar parameters , as well as chemical abundances of the prototype A-type supergiant Deneb to unprecedented accuracy by applying a sophisticated analysis methodology , which has recently been developed and tested .

Methods : The analysis is based on high-S/N and high-resolution spectra in the visual and near-IR .
Stellar parameters and abundances for numerous astrophysically interesting elements are derived from synthesis of the photospheric spectrum using a hybrid non-LTE technique , i.e .
line-blanketed LTE model atmospheres and non-LTE line formation .
Multiple metal ionisation equilibria and numerous hydrogen lines from the Balmer , Paschen , Brackett , and Pfund series are made to match simultaneously for the stellar parameter determination .
The stellar wind properties are derived from H \alpha line-profile fitting using line-blanketed hydrodynamic non-LTE models .
Further constraints come from matching the photospheric spectral energy distribution from the UV to the near-IR L band .

Results : The atmospheric parameters of Deneb are tightly constrained : effective temperature T _ { eff } = 8525 \pm 75 K , surface gravity \log g = 1.10 \pm 0.05 , microturbulence \xi = 8 \pm 1 km s ^ { -1 } , macroturbulence , and projected rotational velocity v \sin i are both 20 \pm 2 km s ^ { -1 } .
The abundance analysis gives helium enrichment by 0.10 dex relative to solar and an N/C ratio of 4.44 \pm 0.84 ( mass fraction ) , implying strong mixing with CN-processed matter .
The heavier elements are consistently underabundant by \sim 0.20 dex compared to solar .
Peculiar abundance patterns , which were derived in previous analyses to exist in Deneb , can not be confirmed .
Accounting for non-LTE effects is essential for removing systematic trends in the abundance determination , for minimising statistical 1 \sigma -uncertainties to \lesssim 10-20 % and for establishing all ionisation equilibria at the same time .

Conclusions : A luminosity of ( 1.96 \pm 0.32 ) \times 10 ^ { 5 } L _ { \odot } , a radius of 203 \pm 17 R _ { \odot } , and a current mass of 19 \pm 4 M _ { \odot } are derived .
Comparison with stellar evolution predictions suggests that Deneb started as a fast-rotating late O-type star with M ^ { ZAMS } \simeq 23 M _ { \odot } on the main sequence and is currently evolving to the red supergiant stage .