Options for Index Defragmentations For Critical Tables
I would not recommend the procedure that you mention in your question. If you did that, the  newly built table would end up suffering the same amount of fragmentation as the original and you would be back to square one.

I would recommend using Ola Hallengren's Index and Maintenance scripts to reduce the fragmentation of your indexes:-
https://ola.hallengren.com/sql-server-index-and-statistics-maintenance.html

Ola's script are tried and tested, I've used them for a while now and have had absolutely no problems with them.

So, taking example H from Ola's page. 

Rebuild or reorganize all indexes with fragmentation on the table Production.Product in the database AdventureWorks:-

EXECUTE dbo.IndexOptimize
@Databases = 'AdventureWorks',
@FragmentationLow = NULL,
@FragmentationMedium = 'INDEX_REORGANIZE',
@FragmentationHigh = 'INDEX_REBUILD_ONLINE',
@FragmentationLevel1 = 5,
@FragmentationLevel2 = 30,
@Indexes = 'AdventureWorks.Production.Product 


This will REORGANIZE all indexes on the Production.Product table with fragmentation > 5% and &lt; 30%. It will then REBUILD_ONLINE all indexes on the table with fragmentation > 30%.

The values are based on Microsoft's recommendation here:-
http://msdn.microsoft.com/en-us/library/ms189858.aspx

REORGANIZE is an online operation, leaving the table available for read-write operations.

REBUILD ONLINE can be run as you have Enterprise Edition. Quoting from BOL:-


  "The ONLINE option allows concurrent user access to the underlying
  table or clustered index data and any associated nonclustered indexes
  during these index operations. For example, while a clustered index is
  being rebuilt by one user, that user and others can continue to update
  and query the underlying data"


But be aware, it should be called REBUILD NEARLY ONLINE as a S-lock or Sch-M lock is held at the end of the operation which can block certain operations. Also online rebuilds cannot be performed on indexes containing the following data types:- image, ntext, and text.

Have a read through of Microsoft's guidelines for online operations:-
http://msdn.microsoft.com/en-us/library/ms190981.aspx

Once you get familiar with the scripts on a DEVELOPMENT system you can look at implementing on your PRODUCTION server. A proper schedule should be setup in order to keep the amount of REORGANIZE/REBUILD that have to be run to a minimum.