The excitons in the orthorhombic phase of the perovskite CH$_3$NH$_3$PbI$_3$ are studied using the effective mass approximation. The electron-hole interaction is screened by a distance-dependent dielectric function, as described by the Haken potential or the Pollmann-B\"uttner potential. The energy spectrum and the eigenfunctions are calculated for both cases. The effective masses, the low and high frequency dielectric constants, and the interband absorption matrix elements, are obtained from generalized density functional theory calculations. The results show that the Pollmann-B\"uttner model provides better agreement with the experimental results. The discrete part of the exciton spectrum is composed of a fundamental state with a binding energy of 24 meV, and higher states that are within 2 meV from the onset the unbound exciton continuum. Light absorption is dominated by the fundamental line with an oscillator strength of 0.013, followed by the exciton continuum. The calculations have been performed without fitting any parameter from experiments and are in close agreement with recent experimental results.