Ok. In the meantime I figured out that Prony seems to be a name.

Do you perhaps mean something like this (ignore the error message that appears - at least in my system)

prony = e0 + Sum[e[i] Exp[-(t/tau[i])], {i, 1, 19}]
param = Flatten[Join[{e0}, Table[{e[i], tau[i]}, {i, 1, 19}]]]
fit = FindFit[data, prony, param, t]
Plot[prony /. fit, {t, 0, 150}, Epilog -> Point /@ data,  PlotRange -> All]

My mistake :

data = {{   3.81E-08  ,    0.202581   }
{   8.22E-08  ,    0.22534    }
{   1.56E-07  ,    0.263415   }
{   1.77E-07  ,    0.27437    }
{   3.36E-07  ,    0.352769   }
{   3.81E-07  ,    0.372474   }
{   7.23E-07  ,    0.521425   }
{   7.31E-07  ,    0.507203   }
{   8.22E-07  ,    0.537803   }
{   1.56E-06  ,    0.732571   }
{   1.57E-06  ,    0.713165   }
{   1.77E-06  ,    0.753371   }
{   3.36E-06  ,    0.999272   }
{   3.39E-06  ,    0.978897   }
{   3.81E-06  ,    1.03047    }
{   7.23E-06  ,    1.35962    }
{   7.31E-06  ,    1.3339 }
{   8.22E-06  ,    1.40366    }
{   1.56E-05  ,    1.85884    }
{   1.57E-05  ,    1.82376    }
{   1.77E-05  ,    1.92504    }
{   3.36E-05  ,    2.57341    }
{   3.39E-05  ,    2.56778    }
{   3.81E-05  ,    2.67567    }
{   4.36E-05  ,    2.85621    }
{   7.23E-05  ,    3.53946    }
{   7.31E-05  ,    3.53409    }
{   8.22E-05  ,    3.67948    }
{   9.39E-05  ,    3.90283    }
{   0.000155861   , 4.77778 }
{   0.000157467   , 4.77227 }
{   0.000177057   , 4.96328 }
{   0.000202191   , 5.25015 }
{   0.000335792   , 6.38345 }
{   0.000339252   , 6.36833 }
{   0.000381459   , 6.58298 }
{   0.000435605   , 6.93619 }
{   0.000723441   , 8.17887 }
{   0.000730896   , 8.14628 }
{   0.000938485   , 8.90897 }
{   0.00155861    ,  10.9395  }
{   0.0015614 ,   10.8011   }
{   0.00157467    ,  10.7765  }
{   0.0020219 ,   11.7196   }
{   0.00336408    ,  13.718   }
{   0.00339252    ,  13.6073  }
{   0.00435605    ,  14.8401  }
{   0.00724748    ,  17.9584  }
{   0.00730896    ,  17.8678  }
{   0.00938484    ,  19.8201  }
{   0.0156141 ,   25.1663   }
{   0.020219  ,    28.8535    }
{   0.0268483 ,   34.0035   }
{   0.0336398 ,   39.0339   }
{   0.0435605 ,   46.0734   }
{   0.0578453 ,   55.7655   }
{   0.0724747 ,   65.2136   }
{   0.0938483 ,   78.5576   }
{   0.124621  ,    96.9397    }
{   0.156142  ,    114.854    }
{   0.20219   , 140.088 }
{   0.268485  ,    174.772    }
{   0.336397  ,    208.272    }
{   0.435605  ,    254.819    }
{   0.578436  ,    317.221    }
{   0.618985  ,    328.147    }
{   0.724746  ,    366.807    }
{   1.2462    ,  528.616  }
{   1.33362   , 538.08  }
{   1.56142   , 592.818 }
{   2.68486   , 817.148 }
{   2.87312   , 823.643 }
{   3.36397   , 895.643 }
{   5.78435   , 1176.53 }
{   6.18991   , 1189.2  }
{   7.24745   , 1283.84 }
{   12.462    ,  1668.42  }
{   13.3358   , 1704.64 }
{   15.6142   , 1836.48 }
{   16.6703   , 1830.4  }
{   16.834    ,  1847.67  }
{   26.8486   , 1964.63 }
{   28.7311   , 1976.94 }
{   35.9166   , 2005.88 }
{   36.2694   , 2017.68 }
{   57.8435   , 2063.7  }
{   61.8991   , 2071.22 }
{   77.3779   , 2090.3  }
{   78.1378   , 2093.55 }
{   124.62    ,  2138.75  }
{   133.358   , 2146.72 }
{   166.705   , 2171.14 }
{   168.342   , 2174.12 }
{   268.486   , 2237.78 }
{   287.31    ,  2248.93  }
{   359.156   , 2282.36 }
{   362.683   , 2285.98 }
{   618.992   , 2377.96 }
{   773.778   , 2407.15 }
{   781.377   , 2406.96 }
{   1333.58   , 2492.99 }
{   1667.05   , 2504.64 }
{   1683.42   , 2504.83 }
{   2873.1    ,  2537.75  }};

This master curve should not decrease on both sides. This was a mistake due to an aggregation of frequency curves.

I need to use Prony series to acquire the coefficents to plot the results in a time domain.

using E(t) = E0 +Ei*exp[-t/pi] (Summation of terms)

Forgive me , but I'm still a little confused with results. How do you acquire the coefficients from this?

pi and Ei ?

Sorry for not knowing enough english: what is a Prony series? And what do you mean by

E(t) = E0 +Ei*exp[-t/pi] (Summation of terms) ?

What is Ei, and what is pi in your problem?

Perhaps you could write down the first 3 or four terms of your series, if possible in Mma-consistent form.

Concerning your new data given above you might want to have a look at

pp = FindFit[data, (a t)/(1 + b t), {a, b}, t]
f3 = (a t)/(1 + b t) /. pp
Plot[f3, {t, 0, 150}, Epilog -> Point@data, PlotRange -> All]

Justin, your datapoints look like the overlay of two spectral lines. Why don't you try to fit them with two gaussians (given below as ffe)?

ffe = a Exp[-b (x - c)^2] + u Exp[-v (x - w)^2];
f2 = ffe /. FindFit[data,  ffe, {{a, 2000}, {b, 1}, {c, 4}, {u, 500}, {v, 1}, {w, 1.5}}, x]

and then

Plot[f2, {x, -2, 5}, Epilog -> Point /@ data]