This final analysis of hadronic and leptonic cross-sections and of 
leptonic forward-backward asymmetries in e$^+$e$^-$ collisions with the 
OPAL detector makes use of the full LEP 1 data sample comprising
$161~\mathrm{pb}^{-1}$ of integrated luminosity and $4.5\times10^6$ 
selected Z$^0$ decays. An interpretation of the data in terms of
contributions  from pure Z$^0$ exchange and from $\gamma$/Z interference 
allows the parameters of the Z$^0$ resonance to be determined in a 
model-independent way. Our results are in good agreement with lepton 
universality and consistent with the vector and axial-vector couplings 
predicted in the Standard Model. A fit to the complete dataset yields the 
fundamental Z$^0$ resonance parameters:
$\mathrm {m_Z} =  (91.1852 \pm 0.0030)$~GeV,
$\Gamma_{\mathrm Z} = (2.4948 \pm 0.0041)$~GeV,
$\sigma_{\mathrm h}^0 = (41.501 \pm 0.055)$~nb,
$R_\ell =  20.823 \pm 0.044$, and
$A_{\mathrm {FB}}^{0,\ell} = 0.0145 \pm 0.0017$.  
Transforming these parameters gives a measurement of the ratio between
the decay width into invisible particles and the width to a single species 
of charged lepton, 
$\Gamma_{\mathrm {inv}}/\Gamma_{\ell \ell} = 5.942 \pm 0.027$.  
Attributing the entire invisible width to neutrino decays and 
assuming the Standard Model couplings for neutrinos, this translates into 
a measurement of the effective number of light neutrino species,
$N_{\nu} =  2.984 \pm  0.013$. Interpreting the data within the context 
of the Standard Model allows the mass of the top quark,
m$_{\mathrm t} = (162 ^{+29}_{-16})$~GeV, to be determined through 
its influence on radiative corrections. Alternatively, utilising the direct 
external measurement of m$_{\mathrm t}$ as an additional constraint leads 
to a measurement of the strong coupling constant and the mass of the Higgs 
boson: $\alpha_{\mathrm s}(m_{\mathrm Z}) = 0.127 \pm 0.005$, and 
$m_{\mathrm H} = (390^{+750}_{-280})$~GeV.