Python 2, 75 69 bytes

from random import*;s=0;j=randrange
while j(12):s=s*9+j(-8,9)
print s

It is trivial to check that the while loop in the middle can generate all integers (albeit biased towards zero). "12" is chosen such that there are roughly half of numbers exceeding ±10⁶.

Older solution:

Python 2, 44 bytes

Based on the Mathematica solution.

from random import*;print int(gauss(0,8**7))

Doesn't really work because Python's float has only finite precision.

Python 2, 75 69 bytes

from random import*;s=0;j=randrange
while j(12):s=s*9+j(-8,9)
print s

It is trivial to check that the while loop in the middle can generate all integers (albeit biased towards zero). "12" is chosen such that there are roughly half of numbers exceeding ±10⁶.

Older solution:

Python 2, 44 bytes

Based on the Mathematica solution.

from random import*;print int(gauss(0,8**7))

Doesn't really work because Python's float has only finite precision.

Python 2, 75 bytes

from random import*;s=0;j=randrange
while j(11):s=s*11+j(11)
print s/2^s%-2

It is trivial to check that the while loop in the middle can generate all integers (albeit biased towards zero). "11" is chosen such that there are roughly half of numbers exceeding 2×10⁶. The final print code transforms the range from nonnegative integers to all integers.

Older solution:

Python 2, 44 bytes

Based on the Mathematica solution.

from random import*;print int(gauss(0,8**7))

Doesn't really work because Python's float has only finite precision.

Python 2, 75 69 bytes

from random import*;s=0;j=randrange
while j(12):s=s*9+j(-8,9)
print s

It is trivial to check that the while loop in the middle can generate all integers (albeit biased towards zero). "12" is chosen such that there are roughly half of numbers exceeding ±10⁶.

Older solution:

Python 2, 44 bytes

Based on the Mathematica solution.

from random import*;print int(gauss(0,8**7))

Doesn't really work because Python's float has only finite precision.

Python 2, 44 bytes

Based on the Mathematica solution.

from random import*;print int(gauss(0,8**7))

Demonstration of the conditions:

>>> f = [int(gauss(0,8**7)) for _ in xrange(10**6)]
>>> sum(1.0 for x in f if x >= 1e6) / len(f)
0.31688
>>> sum(1.0 for x in f if x <= -1e6) / len(f)
0.316475
>>> max(f)
10119326
>>> min(f)
-10028865

Python 2, 75 bytes

from random import*;s=0;j=randrange
while j(11):s=s*11+j(11)
print s/2^s%-2

It is trivial to check that the while loop in the middle can generate all integers (albeit biased towards zero). "11" is chosen such that there are roughly half of numbers exceeding 2×10⁶. The final print code transforms the range from nonnegative integers to all integers.

Older solution:

Python 2, 44 bytes

Based on the Mathematica solution.

from random import*;print int(gauss(0,8**7))

Doesn't really work because Python's float has only finite precision.