There is no any official conversion or comparison between IELTS and TOEFL iBT scores. However, different universities and institutions use their own methods to compare these scores. Most universities compare IELTS 6.0 band score with TOEFL iBT 80 score. Below I’ll give you some IELTS scores and their widely accepted corresponding TOEFL iBT score values:

IELTS – TOEFL iBT
4.0 – 53
4.5 – 60
5.0 – 67
5.5 – 73
6.0 – 80
6.5 – 87
7.0 – 93
7.5 – 100
8.0 – 107
8.5 – 113
9.0 – 120

This is the conversion chart used by most of the institutions to evaluate your score during admission. But, it isn’t a official conversion chart. It is simply calculated by converting IELTS score into percentage and same percentage into TOEFL score. 

As TOEFL iBT and IELTS are two different tests, the comparison between these two cannot be exact. However, on the basis of the research conducted by ETS on more than 1,100 individuals who took both tests, ETS has published following comparison and conversion chart.

Conversion and Comparison between TOEFL iBT and IELTS

TOEFL iBT 35 = IELTS 5.0
TOEFL iBT 46 = IELTS 5.5
TOEFL iBT 60 = IELTS 6.0
TOEFL iBT 79 = IELTS 6.5
TOEFL iBT 94 = IELTS 7.0
TOEFL iBT 102 = IELTS 7.5
TOEFL iBT 110 = IELTS 8.0
TOEFL iBT 115 = IELTS 8.5

I guess the above chart has been extracted from the following link:
http://www.ets.org/s/toefl/pdf/IELTS_Comparison_Chart.pdf

However, I do not agree with the above chart. ETS is trying to show the superiority of TOEFL iBT over IELTS on the basis of so called research conducted by them on more than 1,100 individuals who tool both tests. The comparison is totally misleading. We know that most of the universities in USA have minimum English language proficiency requirements (79 in TOEFL iBT or 6.0 in IELTS) for international students to enter into most of their undergraduate programs. But ESL states 79 in TOEFL iBT is equivalent to 6.5 in IELTS which contradicts with minimum requirements of most of the universities.

I suggest to use the following comparison chart which I personally feel is the correct one.

IELTS to iBT TOEFL Score Conversion

9.0 in IELTS = 120 in iBT TOEFL
8.5 in IELTS = 117 in iBT TOEFL
8.0 in IELTS = 111-112 in iBT TOEFL
7.5 in IELTS = 105-108 in iBT TOEFL
7.0 in IELTS = 100 in iBT TOEFL
6.5 in IELTS = 88-89 in iBT TOEFL
6.0 in IELTS = 79-80 in iBT TOEFL
5.5 in IELTS = 68 in iBT TOEFL
5.0 in IELTS = 61 in iBT TOEFL
4.5 in IELTS = 53 in iBT TOEFL
4.0 in IELTS = 41-42 in iBT TOEFL
3.5 in IELTS = 35-37 in iBT TOEFL
3.0 in IELTS = 30-31 in iBT TOEFL
2.5 in IELTS = 24-25 in iBT TOEFL
2.0 in IELTS = 19-20 in iBT TOEFL
1.5 in IELTS = 15-16 in iBT TOEFL
1.0 in IELTS = 12 in iBT TOEFL

Source: http://gradschool.ufl.edu/pdf-files/english-test-concordance.pdf

Personally, I think the following table is unfair for those who take the TOEFL test.
9.0 in IELTS = 120 in iBT TOEFL
8.5 in IELTS = 117 in iBT TOEFL
8.0 in IELTS = 111-112 in iBT TOEFL
7.5 in IELTS = 105-108 in iBT TOEFL
7.0 in IELTS = 100 in iBT TOEFL
6.5 in IELTS = 88-89 in iBT TOEFL
6.0 in IELTS = 79-80 in iBT TOEFL
5.5 in IELTS = 68 in iBT TOEFL
5.0 in IELTS = 61 in iBT TOEFL
4.5 in IELTS = 53 in iBT TOEFL

I 've learnt both, and find that, if they use such table, everyone will take IELTS, since it is easy to get 6.5 on IELTS but very difficult to get 90 on TOEFL iBT.

Fortunely, following the approval by the Hon. Chris Bowen MP,Minister for Immigration and Citizenship of the TOEFL® test for student visas in May 2011, the Minister has now accepted the test score equivalencies identified by ETS research for the TOEFL test when compared to International English Language Testing System (IELTS®) scores.
The new score equivalencies is
TOEFL iBT® - IELTS
31 - 4.0
32 - 4.5
35 - 5.0
46 -5.5
60 - 6.0
79 - 6.5
94 - 7.0
102 - 7.5
110 - 8.0
115 - 8.5
118 - 9.0
which is what flyAbroad has posted.

Source: http://www.gettingdownunder.com/2011/10/27/toefl-test-score-equivalencies-for-student-visas-in-australia/