Is there any way to connect ethernet to arduino mega board?
Which ethernet shield? There are a couple. If the shield uses SPI the standard libraries probably won't work since the mega has its SPI pins relocated to another place. Your options are either
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hardware/software hack where you physically connect the arduino mega's SPI pins to the SPI pins on the shield and patch the software library (a minor change).
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complete software solution where you do SPI in software. This solution does not use hardware SPI so it's slower than solution 1, but you don't need any (ugly) hardware mods.
I have seen shields advertised as being compatible with the mega, such as http://www.nkcelectronics.com/nkc-ethernet-shield-for-arduino-mega--duemilanove--diecimila-diy-kit.html.