In the February 13, 2012 article,
Some imagine Chester Street for new bridge, Scott Bennett, the director of the Arkansas Highway and Transportation Department, gave a current estimate of $750 million for construction of the Great River Bridge:
The Great River Bridge, a $750 million proposed crossing on the Mississippi River near Arkansas City. It is part of future Interstate 69. It has been designed, but no funding for the bridge has been identified.
The Great River Bridge is also known as the Charles W. Dean Bridge. I-69 Section of Independent Utility 12 ("SIU 12") includes the Charles W. Dean Bridge over the Mississippi River and a short approach from Mississippi. Here's
a map of SIU 12 (which is located primarily in Arkansas). Also,
here are two perspectives of the current design of the Charles W. Dean Bridge:

It should be a very long time before any construction will begin on this bridge; I've started this thread to basically be a repository of information of the occasional snippets of information about cost estimates and other related items. Over time, it should be somewhat similar to the
"I-69 Ohio River Bridge" thread on the Ohio Valley page.
I don't think anyone is even venturing a guess as to when construction might begin. The best recent clue that I have seen is from
page 9/18 of the I-69 Mississippi Section of Independent Utility 11 ("SIU 11") Final Environmental Impact Statement ("FEIS") Executive Summary pdf (page S-9 of the document), in which 2026 is identified as the beginning date for the 22.8 mile section of Section 4 of
SIU 11, which is the SIU 11 section in Mississippi that connects to SIU 12 and the bridge:
Section 4: 22.807 miles, South of SR 446 Interchange to Great River Bridge
Anticipated Letting Date: 2026
Above said, AHTD in the past has informed me that ROW acquisition has already begun for the bridge:
Arkansas 2010-13 STIP, which was released in April, lists a FY 2010 letting for the Charles W. Dean Memorial Bridge over Mississippi River for "Phase I". The project has an earmark of $8.67 million and the total letting is anticipated to be $10.35 million. The recent I-69 book estimates total cost of project to be around $500 million.
http://www.arkansashighways.com/stip/Final%20STIP%202010-2013%20%20Web%20Version.pdf
I-69 Corridor (Hwy. 65-Miss. Rt. 1) (GRB) (Ph I) (F) 069 2010 Phase I $10,346 $8,666
I recently emailed AHTD to see if these projects were still "on track". Pertinent part of the reply:
Phase 1 for the Great River Bridge involves right of way acquisition, which is already underway. Phase I for the Monticello Bypass is preliminary engineering which involves environmental work.
I don’t believe either project is going to make FY 2010.
(above quote from the
"I-69 in AR" thread on Mid-South page)
At any rate, construction of both the I-69 Mississippi River Bridge and the I-69 Ohio River Bridge should give us a new take on the "race of the turtles".
editFor perspective on the $750 million estimate,
a July 25, 2010 article provided a cost estimate of $715 million:
The Interstate 69 bridge’s close proximity to the new U.S. 82 bridge and its price tag — $715 million — raises a question: Why spend money on a new I-69 bridge at Benoit when there is now a completed interstate-grade bridge near Greenville?