Does the accelerated rate of expansion of the Universe have any effect on the speed of light in vacuum? So I was just wondering about this as I finished reading about Michelson-Morley's experiment which disproved the ether theory. My question is since the Universe is constantly expanding  and that too with an accelerated rate does it somehow effect the experimental conditions? If so I wanna know how because I'm unable to understand what effects would expansion of space have on the velocity of light. Is the speed of light not a constant then?
 A: The expansion of the Universe has no effect on the local speed of light. Any local measurement of $c$ will yield $c$, and $c$ won't change.
There is one thing that often causes confusion about the speed of light or faster-than-light travel. A photon moving in an expanding space-time appears to move at an average speed faster than $c$. Consider a simplified/idealized scenario: a photon is directed at a target 2 lightyears away. It travels at $c$ for one year, covering a distance of 1 lightyear. Then the Universe it is travelling in undergoes a near-instantaneous expansion to double its previous size. Now it is 2 light years from its starting point, and 2 lightyears from its target. It keeps travelling at $c$, so it takes 2 more years to reach its target. Now suppose you decided to calculate the average speed of the photon. It is now 4 lightyears from its starting point, and it made the journey in 3 years, so its average speed was $\frac{4}{3}c$. But this is not a local measurement of $c$; I said before that the photon was travelling at $c$ the whole time, and any local measurement at any time would have given $c$. It's just that the space between the origin and target of the photon changed while it travelled. This "sudden" expansion is of course not very realistic and I just chose it to make my point easy to illustrate, but the same concept applies for gradual expansion, demonstrating so just requires a few integrals.
A: The Michelson-Morley experiment indicated (contrarily to it's original intent), that the speed of light perceived by an observer is not dependent on their speed relative to anything.
Along these lines, our movement in space with respect to any arbitrary point has no impact on the speed of light measured in the earth system.
Disclaimer: did not hear a course on general relativity yet.
A: 
Does the accelerated rate of expansion of the Universe have any effect on the velocity of light?

This is the subject of some debate. Check out the Wikipedia Variable Speed of Light article: 
"The idea from Moffat and the team Albrecht–Magueijo is that light propagated as much as 60 orders of magnitude faster in the early universe, thus distant regions of the expanding universe have had time to interact at the beginning of the universe".
I think this idea is wrong myself. See the Einstein digital papers. Einstein said light curves because the speed of light varies with position. He never said light curves because spacetime is curved, that's a popscience myth. See Baez for more: "In that sense, we could say that the 'ceiling' speed of light in the presence of gravity is higher than the 'floor' speed of light". Applying this to cosmology would mean the speed of light was slower in the early universe, not faster.    

So I was just wondering about this as I finished reading about Michelson-Morley's experiment which disproved the ether theory. 

I'm afraid that's popscience too. See the Einstein digital papers for Einstein talking about ether in 1920: "Recapitulating, we may say that according to the general theory of relativity space is endowed with physical qualities; in this sense, therefore, there exists an ether". Einstein removed the need for the luminiferous aether in 1905, but he was still saying space is an aether in 1920. Search arXiv for papers with aether in the title.   

My question is since the Universe is constantly expanding and that too with an accelerated rate does it somehow effect the experimental conditions? If so I wanna know how because I'm unable to understand what effects would expansion of space have on the velocity of light. 

Nobody can give you any definite answers at the moment. In five years maybe the situation will be different. 

Is the speed of light not a constant then?

No. The locally-measured speed of light appears to be constant because we use the local motion of light to define our second and our metre, which we then use to measure the local speed of light! It's a tautology. Magueijo and Moffat referred to it in http://arxiv.org/abs/0705.4507:
"Following Ellis, let us first consider c as the speed of the photon. Can c vary? Could such a variation be measured? As correctly pointed out by Ellis, within the current protocol for measuring time and space the answer is no. The unit of time is defined by an oscillating system or the frequency of an atomic transition, and the unit of space is defined in terms of the distance travelled by light in the unit of time. We therefore have a situation akin to saying that the speed of light is 'one light-year per year', i.e. its constancy has become a tautology or a definition."
